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On July 20, 1969, Armstrong and Apollo 11 Lunar Module (LM) pilot Buzz Aldrin became the first people to land on the Moon, and the next day they spent two and a half hours outside the Lunar Module "Eagle" spacecraft while Michael Collins remained in lunar orbit in the Apollo Command Module "Columbia". When Armstrong fi...
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After he resigned from NASA in 1971, Armstrong taught in the Department of Aerospace Engineering at the University of Cincinnati until 1979. He served on the Apollo 13 accident investigation and on the Rogers Commission, which investigated the Space Shuttle "Challenger" disaster. In 2012, Armstrong died due to complica...
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Armstrong was born near Wapakoneta, Ohio, on August 5, 1930, the son of Viola Louise (née Engel) and Stephen Koenig Armstrong. He was of German, Scots-Irish, and Scottish descent. He is a descendant of Clan Armstrong. He had a younger sister, June, and a younger brother, Dean. His father was an auditor for the Ohio sta...
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The family's last move was in 1944 and took them back to Wapakoneta, where Armstrong attended Blume High School and took flying lessons at the Wapakoneta airfield. He earned a student flight certificate on his 16th birthday, then soloed in August, all before he had a driver's license. He was an active Boy Scout and ear...
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At age 17, in 1947, Armstrong began studying aeronautical engineering at Purdue University in West Lafayette, Indiana; he was the second person in his family to attend college. Armstrong was also accepted to the Massachusetts Institute of Technology (MIT), but he resolved to go to Purdue after watching a football game ...
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Armstrong's call-up from the Navy arrived on January 26, 1949, requiring him to report to Naval Air Station Pensacola in Florida for flight training with class 5-49. After passing the medical examinations, he became a midshipman on February 24, 1949. Flight training was conducted in a North American SNJ trainer, in whi...
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Armstrong was assigned to Fleet Aircraft Service Squadron7 (FASRON 7) at NAS San Diego (now known as NAS North Island). On November 27, 1950, he was assigned to VF-51, an all-jet squadron, becoming its youngest officer, and made his first flight in a jet, a Grumman F9F Panther, on January 5, 1951. He was promoted to en...
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On August 29, 1951, Armstrong saw action in the Korean War as an escort for a photo reconnaissance plane over Songjin. Five days later, on September 3, he flew armed reconnaissance over the primary transportation and storage facilities south of the village of Majon-ni, west of Wonsan. According to Armstrong, he was mak...
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Armstrong flew the plane back to friendly territory, but due to the loss of the aileron, ejection was his only safe option. He intended to eject over water and await rescue by Navy helicopters, but his parachute was blown back over land. A jeep driven by a roommate from flight school picked him up; it is unknown what h...
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In all, Armstrong flew 78missions over Korea for a total of 121hours in the air, a third of them in January 1952, with the final mission on March 5, 1952. Of 492 U.S. Navy personnel killed in the Korean War, 27 of them were from "Essex" on this war cruise. Armstrong received the Air Medal for 20 combat missions, two go...
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Armstrong's regular commission was terminated on February 25, 1952, and he became an ensign in the United States Navy Reserve. On completion of his combat tour with "Essex", he was assigned to a transport squadron, VR-32, in May 1952. He was released from active duty on August 23, 1952, but remained in the reserve, and...
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After his service with the Navy, Armstrong returned to Purdue. His previously earned good but not outstanding grades now improved, lifting his final Grade Point Average (GPA) to a respectable but not outstanding 4.8 out of 6.0. He pledged the Phi Delta Theta fraternity, and lived in its fraternity house. He wrote and c...
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Armstrong met Janet Elizabeth Shearon, who was majoring in home economics, at a party hosted by Alpha Chi Omega. According to the couple, there was no real courtship, and neither could remember the exact circumstances of their engagement. They were married on January 28, 1956, at the Congregational Church in Wilmette, ...
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Following his graduation from Purdue, Armstrong became an experimental research test pilot. He applied at the National Advisory Committee for Aeronautics (NACA) High-Speed Flight Station at Edwards Air Force Base. NACA had no open positions, and forwarded his application to the Lewis Flight Propulsion Laboratory in Cle...
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On his first day, Armstrong was tasked with piloting chase planes during releases of experimental aircraft from modified bombers. He also flew the modified bombers, and on one of these missions had his first flight incident at Edwards. On March 22, 1956, he was in a Boeing B-29 Superfortress, which was to air-drop a Do...
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As they climbed to , the number-four engine stopped and the propeller began windmilling (rotating freely) in the airstream. Hitting the switch that would stop the propeller's spinning, Butchart found it slowed but then started spinning again, this time even faster than the others; if it spun too fast, it would break ap...
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Armstrong served as project pilot on Century Series fighters, including the North American F-100 Super Sabre A and C variants, the McDonnell F-101 Voodoo, the Lockheed F-104 Starfighter, the Republic F-105 Thunderchief and the Convair F-106 Delta Dart. He also flew the Douglas DC-3, Lockheed T-33 Shooting Star, North A...
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Armstrong was involved in several incidents that went down in Edwards folklore or were chronicled in the memoirs of colleagues. During his sixth X-15 flight on April 20, 1962, Armstrong was testing the MH-96 control system when he flew to a height of over (the highest he flew before Gemini 8). He held up the aircraft n...
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Fellow astronaut Michael Collins wrote that of the X-15 pilots Armstrong "had been considered one of the weaker stick-and-rudder men, but the very best when it came to understanding the machine's design and how it operated". Many of the test pilots at Edwards praised Armstrong's engineering ability. Milt Thompson said ...
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On April 24, 1962, Armstrong flew for the only time with Yeager. Their job, flying a T-33, was to evaluate Smith Ranch Dry Lake in Nevada for use as an emergency landing site for the X-15. In his autobiography, Yeager wrote that he knew the lake bed was unsuitable for landings after recent rains, but Armstrong insisted...
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On May 21, 1962, Armstrong was involved in the "Nellis Affair". He was sent in an F-104 to inspect Delamar Dry Lake in southern Nevada, again for emergency landings. He misjudged his altitude and did not realize that the landing gear had not fully extended. As he touched down, the landing gear began to retract; Armstro...
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It took thirty minutes to clear the runway and rig another arresting cable. Armstrong telephoned Edwards and asked for someone to collect him. Milt Thompson was sent in an F-104B, the only two-seater available, but a plane Thompson had never flown. With great difficulty, Thompson made it to Nellis, where a strong cross...
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In June 1958, Armstrong was selected for the U.S. Air Force's Man In Space Soonest program, but the Advanced Research Projects Agency (ARPA) canceled its funding on August 1, 1958, and on November 5, 1958, it was superseded by Project Mercury, a civilian project run by NASA. As a NASA civilian test pilot, Armstrong was...
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In April 1962, NASA sought applications for the second group of NASA astronauts for Project Gemini, a proposed two-man spacecraft. This time, selection was open to qualified civilian test pilots. Armstrong visited the Seattle World's Fair in May 1962 and attended a conference there on space exploration that was co-spon...
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NASA's Director of Flight Crew Operations, Deke Slayton, called Armstrong on September 13, 1962, and asked whether he would be interested in joining the NASA Astronaut Corps as part of what the press dubbed "the New Nine"; without hesitation, Armstrong said yes. The selections were kept secret until three days later, a...
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On February 8, 1965, Armstrong and Elliot See were picked as the backup crew for Gemini 5, with Armstrong as commander, supporting the prime crew of Gordon Cooper and Pete Conrad. The mission's purpose was to practice space rendezvous and to develop procedures and equipment for a seven-day flight, all of which would be...
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The crews for Gemini8 were assigned on September 20, 1965. Under the normal rotation system, the backup crew for one mission became the prime crew for the third mission after, but Slayton designated David Scott as the pilot of Gemini8. Scott was the first member of the third group of astronauts, who was selected on Oct...
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Gemini 8 launched on March 16, 1966. It was the most complex mission yet, with a rendezvous and docking with an uncrewed Agena target vehicle, and the planned second American space walk (EVA) by Scott. The mission was planned to last 75hours and 55orbits. After the Agena lifted off at 10:00:00 EST, the Titan II rocket ...
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A few people in the Astronaut Office, including Walter Cunningham, felt that Armstrong and Scott "had botched their first mission". There was speculation that Armstrong could have salvaged the mission if he had turned on only one of the two RCS rings, saving the other for mission objectives. These criticisms were unfou...
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In Armstrong's final assignment in the Gemini program, he was the back-up Command Pilot for Gemini 11. Having trained for two flights, Armstrong was quite knowledgeable about the systems and took on a teaching role for the rookie backup Pilot, William Anders. The launch was on September 12, 1966, with Conrad and Gordon...
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Following the flight, President Lyndon B. Johnson asked Armstrong and his wife to take part in a 24-day goodwill tour of South America. Also on the tour, which took in 11countries and 14major cities, were Dick Gordon, George Low, their wives, and other government officials. In Paraguay, Armstrong greeted dignitaries in...
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On January 27, 1967—the day of the Apollo 1 fire—Armstrong was in Washington, D.C. with Cooper, Gordon, Lovell and Scott Carpenter for the signing of the United Nations Outer Space Treaty. The astronauts chatted with the assembled dignitaries until 18:45, when Carpenter went to the airport, and the others returned to t...
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On April 5, 1967, the same day the Apollo1 investigation released its final report, Armstrong and 17 other astronauts gathered for a meeting with Slayton. The first thing Slayton said was, "The guys who are going to fly the first lunar missions are the guys in this room." According to Cernan, only Armstrong showed no r...
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The crew was officially assigned on November 20, 1967. For crewmates, Armstrong was assigned Lovell and Aldrin, from Gemini 12. After design and manufacturing delays of the lunar module (LM), Apollo 8 and9 swapped prime and backup crews. Based on the normal crew rotation, Armstrong would command Apollo 11, with one cha...
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To give the astronauts practice piloting the LM on its descent, NASA commissioned Bell Aircraft to build two Lunar Landing Research Vehicles (LLRV), later augmented with three Lunar Landing Training Vehicles (LLTV). Nicknamed the "Flying Bedsteads", they simulated the Moon's one-sixth gravity using a turbofan engine to...
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In addition to the LLRV training, NASA began lunar landing simulator training after Apollo 10 was completed. Aldrin and Armstrong trained for a variety of scenarios that could develop during a real lunar landing. They also received briefings from geologists at NASA.
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After Armstrong served as backup commander for Apollo8, Slayton offered him the post of commander of Apollo 11 on December 23, 1968, as Apollo8 orbited the Moon. According to Armstrong's 2005 biography, Slayton told him that although the planned crew was Commander Armstrong, Lunar Module Pilot Buzz Aldrin, and Command ...
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According to Chris Kraft, a March 1969 meeting among Slayton, George Low, Bob Gilruth, and Kraft determined that Armstrong would be the first person on the Moon, in part because NASA management saw him as a person who did not have a large ego. A press conference on April 14, 1969, gave the design of the LM cabin as the...
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A Saturn V rocket launched Apollo 11 from Launch Complex 39A at the Kennedy Space Center on July 16, 1969, at 13:32:00 UTC (09:32:00 EDT local time). Armstrong's wife Janet and two sons watched from a yacht moored on the Banana River. During the launch, Armstrong's heart rate peaked at 110beats per minute. He found the...
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Apollo 11's objective was to land safely on the Moon, rather than to touch down at a precise location. Three minutes into the lunar descent, Armstrong noted that craters were passing about two seconds too early, which meant the Lunar Module "Eagle" would probably touch down several miles (kilometres) beyond the planned...
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When Armstrong noticed they were heading toward a landing area that seemed unsafe, he took manual control of the LM and attempted to find a safer area. This took longer than expected, and longer than most simulations had taken. For this reason, Mission Control was concerned that the LM was running low on fuel. On landi...
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The landing on the surface of the Moon occurred several seconds after 20:17:40 UTC on July 20, 1969. One of three probes attached to three of the LM's four legs made contact with the surface, a panel light in the LM illuminated, and Aldrin called out, "Contact light." Armstrong shut the engine off and said, "Shutdown."...
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The flight plan called for a crew rest period before leaving the module, but Armstrong asked for this be moved to earlier in the evening, Houston time. When he and Aldrin were ready to go outside, "Eagle" was depressurized, the hatch was opened, and Armstrong made his way down the ladder. At the bottom of the ladder Ar...
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Armstrong prepared his famous epigram on his own. In a post-flight press conference, he said that he chose the words "just prior to leaving the LM." In a 1983 interview in "Esquire" magazine, he explained to George Plimpton: "I always knew there was a good chance of being able to return to Earth, but I thought the chan...
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Recordings of Armstrong's transmission do not provide evidence for the indefinite article "a" before "man", though NASA and Armstrong insisted for years that static obscured it. Armstrong stated he would never make such a mistake, but after repeated listenings to recordings, he eventually conceded he must have dropped ...
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When Armstrong made his proclamation, Voice of America was rebroadcast live by the BBC and many other stations worldwide. An estimated 530million people viewed the event, 20 percent out of a world population of approximately 3.6billion.
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About 19minutes after Armstrong's first step, Aldrin joined him on the surface, becoming the second human to walk on the Moon. They began their tasks of investigating how easily a person could operate on the lunar surface. Armstrong unveiled a plaque commemorating the flight, and with Aldrin, planted the flag of the Un...
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After helping to set up the Early Apollo Scientific Experiment Package, Armstrong went for a walk to what is now known as East Crater, east of the LM, the greatest distance traveled from the LM on the mission. His final task was to remind Aldrin to leave a small package of memorial items to Soviet cosmonauts Yuri Gagar...
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After they re-entered the LM, the hatch was closed and sealed. While preparing for liftoff, Armstrong and Aldrin discovered that, in their bulky space suits, they had broken the ignition switch for the ascent engine; using part of a pen, they pushed in the circuit breaker to start the launch sequence. The "Eagle" then ...
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After being released from an 18-day quarantine to ensure that they had not picked up any infections or diseases from the Moon, the crew was feted across the United States and around the world as part of a 38-day "Giant Leap" tour.
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The tour began on August 13, when the three astronauts spoke and rode in ticker-tape parades in their honor in New York and Chicago, with an estimated six million attendees. On the same evening an official state dinner was held in Los Angeles to celebrate the flight, attended by members of Congress, 44governors, the Ch...
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After the tour Armstrong took part in Bob Hope's 1969 USO show, primarily to Vietnam. In May 1970, Armstrong traveled to the Soviet Union to present a talk at the 13th annual conference of the International Committee on Space Research; after arriving in Leningrad from Poland, he traveled to Moscow where he met Premier ...
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Shortly after Apollo 11, Armstrong stated that he did not plan to fly in space again. He was appointed Deputy Associate Administrator for Aeronautics for the Office of Advanced Research and Technology at ARPA, served in the position for a year, then resigned from it and NASA in 1971. He accepted a teaching position in ...
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At Cincinnati, Armstrong was University Professor of Aerospace Engineering. He took a heavy teaching load, taught core classes, and created two graduate-level classes: aircraft design and experimental flight mechanics. He was considered a good teacher, and a tough grader. His research activities during this time did no...
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In 1970, after an explosion aboard Apollo 13 aborted its lunar landing, Armstrong was part of Edgar Cortright's investigation of the mission. He produced a detailed chronology of the flight. He determined that a 28-volt thermostat switch in an oxygen tank, which was supposed to have been replaced with a 65-volt version...
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In 1986, President Ronald Reagan asked Armstrong to join the Rogers Commission investigating the Space Shuttle "Challenger" disaster. Armstrong was made vice chairman of the commission, and held private interviews with contacts he had developed over the years to help determine the cause of the disaster. He helped limit...
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Armstrong was appointed to a fourteen-member commission by President Reagan to develop a plan for American civilian spaceflight in the 21st century. The commission was chaired by former NASA administrator Dr. Thomas O. Paine, with whom Armstrong had worked during the Apollo program. The group published a book titled "P...
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Armstrong and his wife attended the memorial service for the victims of the Space Shuttle "Columbia" disaster in 2003, at the invitation of President George W. Bush.
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After Armstrong retired from NASA in 1971, he acted as a spokesman for several businesses. The first company to successfully approach him was Chrysler, for whom he appeared in advertising starting in January 1979. Armstrong thought they had a strong engineering division, and they were in financial difficulty. He later ...
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In addition to his duties as a spokesman, he also served on the board of directors of several companies. The first company board Armstrong joined was Gates Learjet, chairing their technical committee. He flew their new and experimental jets and even set a climb and altitude record for business jets. Armstrong became a ...
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In 1985, professional expedition leader Mike Dunn organized a trip to take men he deemed the "greatest explorers" to the North Pole. The group included Armstrong, Edmund Hillary, Hillary's son Peter, Steve Fossett, and Patrick Morrow. They arrived at the Pole on April 6, 1985. Armstrong said he was curious to see what ...
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Armstrong's family described him as a "reluctant American hero". He kept a low profile later in his life, leading to the belief that he was a recluse. Recalling Armstrong's humility, John Glenn, the first American to orbit Earth, told CNN: "[Armstrong] didn't feel that he should be out huckstering himself. He was a hum...
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Andrew Chaikin says in "A Man on the Moon" that Armstrong kept a low profile but was not a recluse, citing his participation in interviews, advertisements for Chrysler, and hosting a cable television series. Between 1991 and 1993, he hosted "First Flights with Neil Armstrong", an aviation history documentary series on ...
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Armstrong guarded the use of his name, image, and famous quote. When it was launched in 1981, MTV wanted to use his quote in its station identification, with the American flag replaced with the MTV logo, but he refused the use of his voice and likeness. He sued Hallmark Cards in 1994, when they used his name, and a rec...
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For many years, he wrote letters congratulating new Eagle Scouts on their accomplishment, but decided to quit the practice in the 1990s because he felt the letters should be written by people who knew the scout. (In 2003, he received 950congratulation requests.) This contributed to the myth of his reclusiveness. Armstr...
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Some former astronauts, including Glenn and Harrison Schmitt, sought political careers after leaving NASA. Armstrong was approached by groups from both the Democratic and Republican parties, but declined the offers. He supported states' rights and opposed the U.S. acting as the "world's policeman".
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When Armstrong applied at a local Methodist church to lead a Boy Scout troop in the late 1950s, he gave his religious affiliation as "deist". His mother later said that his religious views caused her grief and distress in later life, as she was more religious. Upon his return from the Moon, Armstrong gave a speech in f...
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In 1972, Armstrong visited the Scottish town of Langholm, the traditional seat of Clan Armstrong. He was made the first freeman of the burgh, and happily declared the town his home. To entertain the crowd, the Justice of the Peace read from an unrepealed archaic 400-year-old law that required him to hang any Armstrong ...
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Armstrong flew light aircraft for pleasure. He enjoyed gliders and before the moon flight had earned a gold badge with two diamonds from the International Gliding Commission. He continued to fly engineless aircraft well into his 70's.
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While working on his farm in November 1978, Armstrong jumped off the back of his grain truck and caught his wedding ring in its wheel, tearing the tip off his left ring finger. He collected the severed tip, packed it in ice, and had surgeons reattach it at the Jewish Hospital in Louisville, Kentucky. In February 1991, ...
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Armstrong and his first wife, Janet, separated in 1990 and divorced in 1994 after 38 years of marriage. He met his second wife, Carol Held Knight, at a golf tournament in 1992, when they were seated together at breakfast. She said little to Armstrong, but he called her two weeks later to ask what she was doing. She rep...
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In May 2005, Armstrong became involved in a legal dispute with Mark Sizemore, his barber of 20years. After cutting Armstrong's hair, Sizemore sold some of it to a collector for $3,000 without Armstrong's knowledge. Armstrong threatened legal action against Sizemore unless he returned the hair or donated the proceeds to...
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Armstrong underwent bypass surgery at Mercy Faith–Fairfield Hospital in Cincinnati on August 7, 2012, to relieve coronary artery disease. Although he was reportedly recovering well, he developed complications and died on August 25, aged 82. President Barack Obama issued a statement memorializing Armstrong as "among the...
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Armstrong's family released a statement describing him as a "reluctant American hero [who had] served his nation proudly, as a navy fighter pilot, test pilot, and astronaut ... While we mourn the loss of a very good man, we also celebrate his remarkable life and hope that it serves as an example to young people around ...
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Buzz Aldrin called Armstrong "a true American hero and the best pilot I ever knew", and said he was disappointed that they would not be able to celebrate the 50th anniversary of the Moon landing together in 2019. Michael Collins said, "He was the best, and I will miss him terribly." NASA Administrator Charles Bolden sa...
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A tribute was held for Armstrong on September 13, at Washington National Cathedral, whose Space Window depicts the Apollo 11 mission and holds a sliver of Moon rock amid its stained-glass panels. In attendance were Armstrong's Apollo 11 crewmates, Collins and Aldrin; Gene Cernan, the Apollo 17 mission commander and las...
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In July 2019, after observations of the 50th anniversary of the Moon landing, "The New York Times" reported on details of a medical malpractice suit Armstrong's family had filed against Mercy Health–Fairfield Hospital, where he died. When Armstrong appeared to be recovering from his bypass surgery, nurses removed the w...
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The family ultimately settled for $6 million in 2014. Letters included with the 93 pages of documents sent to the "Times" by an unknown individual show that his sons intimated to the hospital, through their lawyers, that they might discuss what happened to their father publicly at the 45th anniversary observances in 20...
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When Pete Conrad of Apollo 12 became the third man to walk on the Moon, on November 19, 1969, his first words referenced Armstrong. The shorter of the two, when Conrad stepped from the LM onto the surface he proclaimed "Whoopie! Man, that may have been a small one for Neil, but that's a long one for me."
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Armstrong received many honors and awards, including the Presidential Medal of Freedom (with distinction) from President Nixon, the Cullum Geographical Medal from the American Geographical Society, and the Collier Trophy from the National Aeronautic Association (1969); the NASA Distinguished Service Medal and the Dr. R...
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Armstrong was elected as member into the National Academy of Engineering in 1978 for contributions to aerospace engineering, scientific knowledge, and exploration of the universe as an experimental test pilot and astronaut. He was elected to the American Philosophical Society in 2001.
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Armstrong and his Apollo 11 crewmates were the 1999 recipients of the Langley Gold Medal from the Smithsonian Institution. On April 18, 2006, he received NASA's Ambassador of Exploration Award. The Space Foundation named Armstrong as a recipient of its 2013 General James E. Hill Lifetime Space Achievement Award. Armstr...
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The lunar crater Armstrong, from the Apollo 11 landing site, and asteroid 6469 Armstrong are named in his honor. There are more than a dozen elementary, middle and high schools named for Armstrong in the United States, and many places around the world have streets, buildings, schools, and other places named for him and...
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In October 2004 Purdue University named its new engineering building Neil Armstrong Hall of Engineering; the building was dedicated on October 27, 2007, during a ceremony at which Armstrong was joined by fourteen other Purdue astronauts. The NASA Dryden Flight Research Center was renamed the NASA Neil A. Armstrong Flig...
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Armstrong's authorized biography, "", was published in 2005. For many years, he turned down biography offers from authors such as Stephen Ambrose and James A. Michener, but agreed to work with James R. Hansen after reading one of Hansen's other biographies. He recalled his initial concerns about the Apollo 11 mission, ...
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In July 2018, Armstrong's sons put his collection of memorabilia up for sale, including his Boy Scout cap, and various flags and medals flown on his space missions. A series of auctions was held on November 1 to 3, 2018, that realized $5,276,320. , the auction sales have totaled $16.7million. Two fragments of wood from...
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Armstrong donated his papers to Purdue. Along with posthumous donations by his widow Carol, the collection consists of over 450boxes of material. In May 2019, she donated two pieces of fabric from the "Wright Flyer", along with his correspondence related to them.
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In a 2010 "Space Foundation" survey, Armstrong was ranked as the most popular space hero; and in 2013, "Flying" magazine ranked him #1 on its list of 51 Heroes of Aviation. The press often asked Armstrong for his views on the future of spaceflight. In 2005, he said that a human mission to Mars would be easier than the ...
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The planetarium at Altoona Area High School in Altoona, Pennsylvania is named after Neil Armstrong and is home to a Space Race museum.
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The number (; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number appears in many formulas across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integ...
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For thousands of years, mathematicians have attempted to extend their understanding of , sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of for practical computations. Around 250BC, the Greek mathemati...
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The invention of calculus soon led to the calculation of hundreds of digits of , enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and computer scientists have pursued new approaches that, when combined with increasing computational power, extended the decima...
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Because its definition relates to the circle, is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses and spheres. It is also found in formulae from other topics in science, such as cosmology, fractals, thermodynamics, mechanics, and electromagnetism. In modern mathematical...
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The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase Greek letter, sometimes spelled out as "pi." In English, is pronounced as "pie" ( ). In mathematical use, the lowercase letter is distinguished from its capitalized and enlarged counterpart , which deno...
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The ratio is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle, it will also have twice the circumference, preserving the ratio . This definition of implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curve ...
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Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined independently of geometry using limits—a concept in calculus. For example, one may directly compute the arc length of the top half of the unit circle, given in Cartesian coordinates by the ...
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An integral such as this was adopted as the definition of by Karl Weierstrass, who defined it directly as an integral in 1841.
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Integration is no longer commonly used in a first analytical definition because, as explains, differential calculus typically precedes integral calculus in the university curriculum, so it is desirable to have a definition of that does not rely on the latter. One such definition, due to Richard Baltzer and popularized ...
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In a similar spirit, can be defined using properties of the complex exponential, , of a complex variable . Like the cosine, the complex exponential can be defined in one of several ways. The set of complex numbers at which is equal to one is then an (imaginary) arithmetic progression of the form:
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A variation on the same idea, making use of sophisticated mathematical concepts of topology and algebra, is the following theorem: there is a unique (up to automorphism) continuous isomorphism from the group R/Z of real numbers under addition modulo integers (the circle group), onto the multiplicative group of complex ...
https://en.wikipedia.org/wiki?curid=23601