text:
August 2

Pre-1600

 * 338 BC – A Macedonian army led by Philip II defeated the combined forces of Athens and Thebes in the Battle of Chaeronea, securing Macedonian hegemony in Greece and the Aegean.
 * 216 BC – The Carthaginian army led by Hannibal defeats a numerically superior Roman army at the Battle of Cannae.
 * 49 BC – Caesar, who marched to Spain earlier in the year leaving Marcus Antonius in charge of Italy, defeats Pompey's general Afranius and Petreius in Ilerda (Lerida) north of the Ebro river.
 * 461 – Majorian is arrested near Tortona (northern Italy) and deposed by the Suebian general Ricimer as puppet emperor.
 * 932 – After a two-year siege, the city of Toledo, in Spain, surrenders to the forces of the Caliph of Córdoba Abd al-Rahman III, assuming an important victory in his campaign to subjugate the Central March.
 * 1274 – Edward I of England returns from the Ninth Crusade and is crowned King seventeen days later.
 * 1343 – After the execution of her husband, Jeanne de Clisson sells her estates and raises a force of men with which to attack French shipping and ports.
 * 1377 – Russian troops are defeated by forces of the Blue Horde Khan Arapsha in the Battle on Pyana River.
 * 1415 – Thomas Grey is executed for participating in the Southampton Plot.
 * 1492 – The Jews are expelled from Spain: 40,000–200,000 leave. Sultan Bayezid II of the Ottoman Empire, learning of this, dispatches the Ottoman Navy to bring the Jews safely to Ottoman lands, mainly to the cities of Thessaloniki (in modern-day Greece) and İzmir (in modern-day Turkey).

1601–1900

 * 1610 – During Henry Hudson's search for the Northwest Passage, he sails into what is now known as Hudson Bay.
 * 1776 – The signing of the United States Declaration of Independence took place.
 * 1790 – The first United States Census is conducted.
 * 1798 – French Revolutionary Wars: The Battle of the Nile concludes in a British victory.
 * 1830 – Charles X of France abdicates the throne in favor of his grandson Henri.
 * 1858 – The Government of India Act 1858 replaces Company rule in India with that of the British Raj.
 * 1869 – Japan's Edo society class system is abolished as part of the Meiji Restoration reforms.
 * 1870 – Tower Subway, the world's first underground tube railway, opens in London, England, United Kingdom.
 * 1873 – The Clay Street Hill Railroad begins operating the first cable car in San Francisco's famous cable car system.
 * 1897 – Anglo-Afghan War: The Siege of Malakand ends when a relief column is able to reach the British garrison in the Malakand states.

1901–present

 * 1903 – The Ilinden–Preobrazhenie Uprising against the Ottoman Empire begins.
 * 1914 – The German occupation of Luxembourg during World War I begins.
 * 1916 – World War I: Austrian sabotage causes the sinking of the Italian battleship Leonardo da Vinci in Taranto.
 * 1918 – The first general strike in Canadian history takes place in Vancouver.
 * 1922 – A typhoon hits Shantou, Republic of China, killing more than 50,000 people.
 * 1923 – Vice President Calvin Coolidge becomes U.S. President upon the death of President Warren G. Harding.
 * 1932 – The positron (antiparticle of the electron) is discovered by Carl D. Anderson.
 * 1934 – Reichskanzler Adolf Hitler becomes Führer of Germany following the death of President Paul von Hindenburg.
 * 1937 – The Marihuana Tax Act of 1937 is passed in America, the effect of which is to render marijuana and all its by-products illegal.
 * 1939 – Albert Einstein and Leo Szilard write a letter to Franklin D. Roosevelt, urging him to begin the Manhattan Project to develop a nuclear weapon.
 * 1943 – The Holocaust: Jewish prisoners stage a revolt at Treblinka, one of the deadliest of Nazi death camps where approximately 900,000 persons were murdered in less than 18 months.
 * 1943  – World War II: The Motor Torpedo Boat PT-109 is rammed by the Japanese destroyer Amagiri and sinks. Lt. John F. Kennedy, future U.S. president, saves all but two of his crew.
 * 1944 – ASNOM: Birth of the Socialist Republic of Macedonia, celebrated as Day of the Republic in North Macedonia.
 * 1944  – World War II: The largest trade convoy of the world wars arrives safely in the Western Approaches.
 * 1945 – World War II: End of the Potsdam Conference.
 * 1947 – A British South American Airways Avro Lancastrian airliner crashes into a mountain during a flight from Buenos Aires, Argentina to Santiago, Chile. The wreckage would not be found until 1998.
 * 1968 – An earthquake hits Casiguran, Aurora, Philippines killing more than 270 people and wounding 261.
 * 1973 – A flash fire kills 50 people at the Summerland amusement centre at Douglas, Isle of Man.
 * 1980 – A bomb explodes at the railway station in Bologna, Italy, killing 85 people and wounding more than 200.
 * 1982 – The Helsinki Metro, the first rapid transit system of Finland, is opened to the general public.
 * 1985 – Delta Air Lines Flight 191, a Lockheed L-1011 TriStar, crashes at Dallas/Fort Worth International Airport killing 137.
 * 1989 – Pakistan is re-admitted to the Commonwealth of Nations after having restored democracy for the first time since 1972.
 * 1989  – A massacre is carried out by an Indian Peace Keeping Force in Sri Lanka killing 64 ethnic Tamil civilians.
 * 1990 – Iraq invades Kuwait, eventually leading to the Gulf War.
 * 1999 – The Gaisal train disaster claims 285 lives in Assam, India.
 * 2005 – Air France Flight 358 lands at Toronto Pearson International Airport and runs off the runway, causing the plane to burst into flames leaving 12 injuries and no fatalities.
 * 2014 – At least 146 people were killed and more than 114 injured in a factory explosion in Kunshan, Jiangsu, China.

Pre-1600

 * 1260 – Kyawswa of Pagan, last ruler of the Pagan Kingdom (d. 1299)
 * 1455 – John Cicero, Elector of Brandenburg (d. 1499)
 * 1533 – Theodor Zwinger, Swiss physician and scholar (d. 1588)
 * 1549 – Mikołaj Krzysztof "the Orphan" Radziwiłł, Polish nobleman (d. 1616)

1601–1900

 * 1612 – Saskia van Uylenburgh, Dutch model and wife of Rembrandt van Rijn (d. 1642)
 * 1627 – Samuel Dirksz van Hoogstraten, Dutch painter (d. 1678)
 * 1630 – Estephan El Douaihy, Maronite patriarch (d. 1704)
 * 1646 – Jean-Baptiste du Casse, French admiral and buccaneer (d. 1715)
 * 1672 – Johann Jakob Scheuchzer, Swiss paleontologist and scholar (d. 1733)
 * 1674 – Philippe II, Duke of Orléans (d. 1723)
 * 1696 – Mahmud I, Ottoman sultan (d. 1754)
 * 1702 – Dietrich of Anhalt-Dessau (d. 1769)
 * 1703 – Lorenzo Ricci, Italian religious leader, 18th Superior General of the Society of Jesus (d. 1775)
 * 1740 – Jean Baptiste Camille Canclaux, French general (d. 1817)
 * 1754 – Pierre Charles L'Enfant, French-American architect and engineer, designed Washington, D.C. (d. 1825)
 * 1788 – Leopold Gmelin, German chemist and academic (d. 1853)
 * 1815 – Adolf Friedrich von Schack, German poet and historian (d. 1894)
 * 1820 – John Tyndall, Irish-English physicist and mountaineer (d. 1893)
 * 1828 – Manuel Pavía y Rodríguez de Alburquerque, Spanish general (d. 1895)
 * 1834 – Frédéric Auguste Bartholdi, French sculptor, designed the Statue of Liberty (d. 1904)
 * 1835 – Elisha Gray, American businessman, co-founded Western Electric (d. 1901)
 * 1861 – Prafulla Chandra Ray, Indian chemist and academic (d. 1944)
 * 1865 – Irving Babbitt, American academic and critic (d. 1933)
 * 1865  – John Radecki, Australian stained glass artist (d. 1955)
 * 1867 – Ernest Dowson, English poet, novelist, and short story writer (d. 1900)
 * 1868 – Constantine I of Greece (d. 1923)
 * 1870 – Marianne Weber, German sociologist and suffragist (d. 1954)
 * 1871 – John French Sloan, American painter and illustrator (d. 1951)
 * 1872 – George E. Stewart, Australian-American colonel, Medal of Honor recipient (d. 1946)
 * 1876 – Pingali Venkayya, Indian geologist, designed the Flag of India (d. 1963)
 * 1877 – Ravishankar Shukla, Indian lawyer and politician, 1st Chief Minister of Madhya Pradesh (d. 1956)
 * 1878 – Aino Kallas, Finnish-Estonian author (d. 1956)
 * 1880 – Arthur Dove, American painter and educator (d. 1946)
 * 1882 – Red Ames, American baseball player and manager (d. 1936)
 * 1882  – Albert Bloch, American painter and academic (d. 1961)
 * 1884 – Rómulo Gallegos, Venezuelan author and politician, 46th President of Venezuela (d. 1969)
 * 1886 – John Alexander Douglas McCurdy, Canadian pilot and politician, 20th Lieutenant Governor of Nova Scotia (d. 1961)
 * 1887 – Oskar Anderson, Bulgarian-German mathematician and statistician (d. 1960)
 * 1889 – Margaret Lawrence, American stage actress (d. 1929)
 * 1891 – Arthur Bliss, English composer and conductor (d. 1975)
 * 1891  – Viktor Zhirmunsky, Russian linguist and historian (d. 1971)
 * 1892 – Jack L. Warner, Canadian-born American production manager and producer, co-founded Warner Bros. (d. 1978)
 * 1894 – Bertha Lutz, Brazilian feminist and scientist (d. 1976)
 * 1895 – Matt Henderson, New Zealand cricketer (d. 1970)
 * 1897 – Karl-Otto Koch, German SS officer (d. 1945)
 * 1897  – Max Weber, Swiss lawyer and politician (d. 1974)
 * 1898 – Ernő Nagy, Hungarian fencer (d. 1977)
 * 1899 – Charles Bennett, English director and screenwriter (d. 1995)
 * 1900 – Holling C. Holling, American author and illustrator (d. 1973)
 * 1900  – Helen Morgan, American actress and singer (d. 1941)

1901–present

 * 1902 – Pope Cyril VI of Alexandria (d. 1971)
 * 1902  – Mina Rees, American mathematician (d. 1997)
 * 1905 – Karl Amadeus Hartmann, German composer (d. 1963)
 * 1905  – Myrna Loy, American actress (d. 1993)
 * 1905  – Ruth Nelson, American actress (d. 1992)
 * 1907 – Mary Hamman, American journalist and author (d. 1984)
 * 1910 – Roger MacDougall, Scottish director, playwright, and screenwriter (d. 1993)
 * 1911 – Ann Dvorak, American actress (d. 1979)
 * 1912 – Palle Huld, Danish actor (d. 2010)
 * 1912  – Håkon Stenstadvold, Norwegian painter, illustrator, and critic (d. 1977)
 * 1912  – Vladimir Žerjavić, Croatian economist and author (d. 2001)
 * 1913 – Xavier Thaninayagam, Sri Lankan scholar and academic (d. 1980)
 * 1914 – Félix Leclerc, Canadian singer-songwriter, actor, and poet (d. 1988)
 * 1914  – Big Walter Price, American singer-songwriter and pianist (d. 2012)
 * 1914  – Beatrice Straight, American actress (d. 2001)
 * 1915 – Gary Merrill, American actor (d. 1990)
 * 1916 – Alfonso A. Ossorio, Filipino-American painter and sculptor (d. 1990)
 * 1917 – Wah Chang, Chinese-American artist and designer (d. 2003)
 * 1919 – Nehemiah Persoff, Israeli-American actor (d. 2022)
 * 1920 – Louis Pauwels, French journalist and author (d. 1997)
 * 1920  – Augustus Rowe, Canadian physician and politician (d. 2013)
 * 1921 – Alan Whicker, Egyptian-English journalist (d. 2013)
 * 1922 – Betsy Bloomingdale, American philanthropist and socialite (d. 2016)
 * 1922  – Geoffrey Dutton, Australian historian and author (d. 1998)
 * 1923 – Shimon Peres, Polish-Israeli lawyer and politician, 9th President of Israel (d. 2016)
 * 1923  – Ike Williams, American boxer (d. 1994)
 * 1924 – James Baldwin, American novelist, poet, and critic (d. 1987)
 * 1924  – Joe Harnell, American pianist and composer (d. 2005)
 * 1924  – Carroll O'Connor, American actor, director, producer, and screenwriter (d. 2001)
 * 1925 – K. Arulanandan, Ceylon-American engineer and academic (d. 2004)
 * 1925  – John Dexter, English director and producer (d. 1990)
 * 1925  – John McCormack, Canadian ice hockey player (d. 2017)
 * 1925  – Jorge Rafael Videla, Argentinian general and politician, 43rd President of Argentina (d. 2013)
 * 1927 – Peter Swinnerton-Dyer, English mathematician and academic (d. 2018)
 * 1928 – Malcolm Hilton, English cricketer (d. 1990)
 * 1929 – Roy Crimmins, English trombonist and composer (d. 2014)
 * 1929  – John Gale, English director and producer
 * 1929  – Vidya Charan Shukla, Indian politician, Indian Minister of External Affairs (d. 2013)
 * 1929  – David Waddington, Baron Waddington, English lawyer and politician, Governor of Bermuda (d. 2017)
 * 1930 – Vali Myers, Australian painter and dancer (d. 2003)
 * 1931 – Pierre DuMaine, American bishop and academic (d. 2019)
 * 1931  – Eddie Fuller, South African cricketer (d. 2008)
 * 1931  – Karl Miller, English journalist and critic (d. 2014)
 * 1931  – Viliam Schrojf, Czech footballer (d. 2007)
 * 1932 – Lamar Hunt, American businessman, co-founded the American Football League and World Championship Tennis (d. 2006)
 * 1932  – Peter O'Toole, British-Irish actor and producer (d. 2013)
 * 1933 – Ioannis Varvitsiotis, Greek politician, Greek Minister of Defence
 * 1934 – Valery Bykovsky, Russian general and cosmonaut (d. 2019)
 * 1935 – Hank Cochran, American singer-songwriter and guitarist (d. 2010)
 * 1936 – Anthony Payne, English composer and author (d. 2021)
 * 1937 – Ron Brierley, New Zealand businessman
 * 1937  – Billy Cannon, American football player and dentist (d. 2018)
 * 1937  – María Duval, Mexican actress and singer
 * 1937  – Garth Hudson, Canadian keyboard player, songwriter, and producer
 * 1938 – Dave Balon, Canadian ice hockey player and coach (d. 2007)
 * 1938  – Pierre de Bané, Israeli-Canadian lawyer and politician (d. 2019)
 * 1938  – Terry Peck, Falkland Islander soldier (d. 2006)
 * 1939 – Benjamin Barber, American theorist, author, and academic (d. 2017)
 * 1939  – Wes Craven, American director, producer, and screenwriter (d. 2015)
 * 1939  – John W. Snow, American businessman and politician, 73rd United States Secretary of the Treasury
 * 1940 – Angel Lagdameo, Filipino archbishop (d. 2022)
 * 1940  – Beko Ransome-Kuti, Nigerian physician and activist (d. 2006)
 * 1940  – Will Tura, Belgian singer-songwriter and guitarist
 * 1941 – Doris Coley, American singer (d. 2000)
 * 1941  – Jules A. Hoffmann, Luxembourgian-French biologist and academic, Nobel Prize laureate
 * 1941  – François Weyergans, Belgian director and screenwriter (d. 2022)
 * 1942 – Isabel Allende, Chilean-American novelist, essayist, essayist
 * 1942  – Leo Beenhakker, Dutch football manager
 * 1942  – Juan Formell, Cuban singer-songwriter and bass player (d. 2014)
 * 1942  – Nell Irvin Painter, American author and historian
 * 1943 – Herbert M. Allison, American lieutenant and businessman (d. 2013)
 * 1943  – Tom Burgmeier, American baseball player and coach
 * 1943  – Jon R. Cavaiani, English-American sergeant, Medal of Honor recipient (d. 2014)
 * 1943  – Rose Tremain, English novelist and short story writer
 * 1944 – Jim Capaldi, English drummer and singer-songwriter (d. 2005)
 * 1944  – Naná Vasconcelos, Brazilian singer and berimbau player (d. 2016)
 * 1945 – Joanna Cassidy, American actress
 * 1945  – Alex Jesaulenko, Austrian-Australian footballer and coach
 * 1945  – Bunker Roy, Indian educator and activist
 * 1945  – Eric Simms, Australian rugby league player and coach
 * 1946 – James Howe, American journalist and author
 * 1947 – Ruth Bakke, Norwegian organist and composer
 * 1947  – Lawrence Wright, American journalist, author, and screenwriter
 * 1948 – Andy Fairweather Low, Welsh singer-songwriter, guitarist, and producer
 * 1948  – Dennis Prager, American radio host and author
 * 1948  – Tapan Kumar Sarkar, Indian-American electrical engineer and academic (d. 2021)
 * 1948  – James Street, American football and baseball player (d. 2013)
 * 1948  – Snoo Wilson, English playwright and screenwriter (d. 2013)
 * 1949 – James Fallows, American journalist and author
 * 1949  – Bertalan Farkas, Hungarian general and cosmonaut
 * 1950 – Jussi Adler-Olsen, Danish author and publisher
 * 1950  – Ted Turner, British guitarist
 * 1951 – Andrew Gold, American singer-songwriter and producer (d. 2011)
 * 1951  – Steve Hillage, English singer-songwriter and guitarist
 * 1951  – Burgess Owens, American football player and politician
 * 1951  – Joe Lynn Turner, American singer-songwriter and guitarist
 * 1951  – Per Westerberg, Swedish businessman and politician, Speaker of the Parliament of Sweden
 * 1952 – Alain Giresse, French footballer and manager
 * 1953 – Donnie Munro, Scottish singer and guitarist
 * 1953  – Butch Patrick, American actor
 * 1953  – Anthony Seldon, English historian and author
 * 1954 – Sammy McIlroy, Northern Irish footballer and manager
 * 1955 – Caleb Carr, American historian and author (d. 2024)
 * 1955  – Tony Godden, English footballer and manager
 * 1955  – Butch Vig, American drummer, songwriter, and record producer
 * 1956 – Fulvio Melia, Italian-American physicist, astrophysicist, and author
 * 1957 – Jacky Rosen, United States senator
 * 1959 – Victoria Jackson, American actress and singer
 * 1959  – Johnny Kemp, Bahamian singer-songwriter and producer (d. 2015)
 * 1959  – Apollonia Kotero, American singer and actress
 * 1960 – Linda Fratianne, American figure skater
 * 1960  – Neal Morse, American singer and keyboard player
 * 1960  – David Yow, American singer-songwriter
 * 1961 – Pete de Freitas, Trinidadian-British drummer and producer (d. 1989)
 * 1962 – Lee Mavers, English singer, songwriter and guitarist
 * 1962  – Cynthia Stevenson, American actress
 * 1963 – Laura Bennett, American architect and fashion designer
 * 1963  – Uğur Tütüneker, Turkish footballer and manager
 * 1964 – Frank Biela, German race car driver
 * 1964  – Mary-Louise Parker, American actress
 * 1965 – Joe Hockey, Australian lawyer and politician, 38th Treasurer of Australia
 * 1965  – Hisanobu Watanabe, Japanese baseball player and coach
 * 1966 – Takashi Iizuka, Japanese wrestler
 * 1966  – Grainne Leahy, Irish cricketer
 * 1966  – Tim Wakefield, American baseball player and sportscaster (d. 2023)
 * 1967 – Aaron Krickstein, American tennis player
 * 1967  – Aline Brosh McKenna, American screenwriter and producer
 * 1968 – Stefan Effenberg, German footballer and sportscaster
 * 1969 – Cedric Ceballos, American basketball player
 * 1969  – Fernando Couto, Portuguese footballer and manager
 * 1970 – Tony Amonte, American ice hockey player and coach
 * 1970  – Kevin Smith, American actor, director, producer, and screenwriter
 * 1970  – Philo Wallace, Barbadian cricketer
 * 1971 – Jason Bell, Australian rugby league player
 * 1971  – Michael Hughes, Irish footballer and manager
 * 1972 – Mohamed Al-Deayea, Saudi Arabian footballer
 * 1972  – Muriel Bowser, American politician, Mayor of Washington, D.C.
 * 1973 – Danie Keulder, Namibian cricketer
 * 1973  – Miguel Mendonca, Zimbabwean journalist and author
 * 1973  – Susie O'Neill, Australian swimmer
 * 1974 – Phil Williams, English journalist and radio host
 * 1975 – Mineiro, Brazilian footballer
 * 1975  – Xu Huaiwen, Chinese-German badminton player and coach
 * 1975  – Tamás Molnár, Hungarian water polo player
 * 1976 – Reyes Estévez, Spanish runner
 * 1976  – Jay Heaps, American soccer player and coach
 * 1976  – Michael Weiss, American figure skater
 * 1976  – Sam Worthington, English-Australian actor and producer
 * 1976  – Mohammad Zahid, Pakistani cricketer
 * 1977 – Edward Furlong, American actor
 * 1978 – Goran Gavrančić, Serbian footballer
 * 1978  – Matt Guerrier, American baseball player
 * 1978  – Deividas Šemberas, Lithuanian footballer
 * 1978  – Dragan Vukmir, Serbian footballer
 * 1979 – Marco Bonura, Italian footballer
 * 1979  – Reuben Kosgei, Kenyan runner
 * 1980 – Ivica Banović, Croatian footballer
 * 1981 – Alexander Emelianenko, Russian mixed martial artist and boxer
 * 1981  – Tim Murtagh, Irish-English cricketer
 * 1982 – Hélder Postiga, Portuguese footballer
 * 1982  – Kerry Rhodes, American football player
 * 1982  – Grady Sizemore, American baseball player
 * 1983 – Michel Bastos, Brazilian footballer
 * 1983  – Huston Street, American baseball player
 * 1984 – Giampaolo Pazzini, Italian footballer
 * 1985 – Stephen Ferris, Irish rugby player
 * 1985  – David Hart Smith, Canadian wrestler
 * 1985  – Britt Nicole, American Christian pop artist
 * 1986 – Mathieu Razanakolona, Canadian skier
 * 1986 – Lily Gladstone, American actress
 * 1988 – Rob Kwiet, Canadian ice hockey player
 * 1988  – Golden Tate, American football player
 * 1989 – Nacer Chadli, Belgian footballer
 * 1990 – Ima Bohush, Belarusian tennis player
 * 1990  – Vitalia Diatchenko, Russian tennis player
 * 1990  – Skylar Diggins-Smith, American basketball player
 * 1992 – Charli XCX, English singer-songwriter
 * 1993 – Gael Bussa, Congolese politician
 * 1994 – Laura Pigossi, Brazilian tennis player
 * 1994  – Laremy Tunsil, American football player
 * 1995 – Kristaps Porziņģis, Latvian basketball player
 * 1996 – Keston Hiura, American baseball player
 * 1996  – Simone Manuel, American swimmer
 * 1997 – Austin Theory, American wrestler
 * 1999 – Mark Lee, Korean-Canadian singer
 * 2000 – Varvara Gracheva, Russian tennis player
 * 2000 – Mohammed Kudus, Ghanaian footballer

Pre-1600

 * 216 BC – Gnaeus Servilius Geminus, Roman consul
 * 216 BC – Lucius Aemilius Paullus, Roman consul and general
 * 216 BC – Marcus Minucius Rufus, Roman consul
 * 257 – Pope Stephen I
 * 575 – Ahudemmeh, Syriac Orthodox Grand Metropolitan of the East.
 * 640 – Pope Severinus
 * 686 – Pope John V
 * 855 – Ahmad ibn Hanbal, Arab theologian and jurist (b. 780)
 * 924 – Ælfweard of Wessex (b. 904)
 * 1075 – Patriarch John VIII of Constantinople
 * 1100 – William II of England (b. 1056)
 * 1222 – Raymond VI, Count of Toulouse (b. 1156)
 * 1277 – Mu'in al-Din Sulaiman Pervane, Chancellor and Regent of the Sultanate of Rum
 * 1316 – Louis of Burgundy (b. 1297)
 * 1330 – Yolande of Dreux, Queen consort of Scotland and Duchess consort of Brittany (b. 1263)
 * 1332 – King Christopher II of Denmark (b. 1276)
 * 1415 – Thomas Grey, English conspirator (b. 1384)
 * 1445 – Oswald von Wolkenstein, Austrian poet and composer (b. 1376)
 * 1451 – Elizabeth of Görlitz (b. 1390)
 * 1511 – Andrew Barton, Scottish admiral (b. 1466)
 * 1512 – Alessandro Achillini, Italian physician and philosopher (b. 1463)
 * 1589 – Henry III of France (b. 1551)

1601–1900

 * 1605 – Richard Leveson, English admiral (b. c. 1570)
 * 1611 – Katō Kiyomasa, Japanese daimyō (b. 1562)
 * 1667 – Francesco Borromini, Swiss architect, designed San Carlo alle Quattro Fontane and Sant'Agnese in Agone (b. 1599)
 * 1696 – Robert Campbell of Glenlyon (b. 1630)
 * 1769 – Daniel Finch, 8th Earl of Winchilsea, English politician, Lord President of the Council (b. 1689)
 * 1788 – Thomas Gainsborough, English painter (b. 1727)
 * 1799 – Jacques-Étienne Montgolfier, French inventor, co-invented the hot air balloon (b. 1745)
 * 1815 – Guillaume Brune, French general and politician (b. 1763)
 * 1823 – Lazare Carnot, French mathematician, general, and politician, president of the National Convention (b. 1753)
 * 1834 – Harriet Arbuthnot, English diarist (b. 1793)
 * 1849 – Muhammad Ali of Egypt, Ottoman Albanian commander (b. 1769)
 * 1854 – Heinrich Clauren, German author (b. 1771)
 * 1859 – Horace Mann, American educator and politician (b. 1796)
 * 1876 – "Wild Bill" Hickok, American sheriff (b. 1837)
 * 1889 – Eduardo Gutiérrez, Argentinian author (b. 1851)
 * 1890 – Louise-Victorine Ackermann, French poet and author (b. 1813)

1901–present

 * 1903 – Eduard Magnus Jakobson, Estonian missionary and engraver (b. 1847)
 * 1903  – Edmond Nocard, French veterinarian and microbiologist (b. 1850)
 * 1911 – Ioryi Mucitano, Aromanian revolutionary
 * 1913 – Ferenc Pfaff, Hungarian architect and academic, designed Zagreb Central Station (b. 1851)
 * 1915 – John Downer, Australian politician, 16th premier of South Australia (b. 1843)
 * 1917 – Jaan Mahlapuu, Estonian military pilot (b. 1894)
 * 1921 – Enrico Caruso, Italian tenor and actor (b. 1873)
 * 1922 – Alexander Graham Bell, Scottish-Canadian engineer, invented the telephone (b. 1847)
 * 1923 – Warren G. Harding, American journalist and politician, 29th president of the United States (b. 1865)
 * 1923 – Joseph Whitty, Irish Republican died on hunger strike during the 1923 Irish Hunger Strikes (b. 1904)
 * 1934 – Paul von Hindenburg, German field marshal and politician, 2nd president of Germany (b. 1847)
 * 1937 – Artur Sirk, Estonian soldier, lawyer, and politician (b. 1900)
 * 1939 – Harvey Spencer Lewis, American mystic and author (b. 1883)
 * 1945 – Pietro Mascagni, Italian composer and educator (b. 1863)
 * 1955 – Alfred Lépine, Canadian ice hockey player and coach (b. 1901)
 * 1955  – Wallace Stevens, American poet and educator (b. 1879)
 * 1963 – Oliver La Farge, American anthropologist and author (b. 1901)
 * 1967 – Walter Terence Stace, English-American epistemologist, philosopher, and academic (b. 1886)
 * 1970 – Angus MacFarlane-Grieve, English academic, mathematician, rower, and soldier (b. 1891)
 * 1972 – Brian Cole, American bass player (b. 1942)
 * 1972  – Paul Goodman, American psychotherapist and author (b. 1911)
 * 1972  – Helen Hoyt, American poet and author (b. 1887)
 * 1973 – Ismail Abdul Rahman, Former Deputy Prime Minister of Malaysia (b. 1915)
 * 1973 – Jean-Pierre Melville, French actor, director, producer, and screenwriter (b. 1917)
 * 1974 – Douglas Hawkes, English race car driver and businessman (b. 1893)
 * 1976 – László Kalmár, Hungarian mathematician and academic (b. 1905)
 * 1976  – Fritz Lang, Austrian-American director, producer, and screenwriter (b. 1890)
 * 1978 – Carlos Chávez, Mexican composer and conductor (b. 1899)
 * 1978  – Antony Noghès, French businessman, founded the Monaco Grand Prix (b. 1890)
 * 1979 – Thurman Munson, American baseball player (b. 1947)
 * 1981 – Kieran Doherty, Irish hunger striker and politician (b. 1955)
 * 1981 – Stefanie Clausen, Danish diver (b. 1900)
 * 1983 – James Jamerson, American bass player (b. 1936)
 * 1986 – Roy Cohn, American lawyer and politician (b. 1927)
 * 1988 – Joe Carcione, American activist and author (b. 1914)
 * 1988  – Raymond Carver, American short story writer and poet (b. 1938)
 * 1990 – Norman Maclean, American short story writer and essayist (b. 1902)
 * 1990  – Edwin Richfield, English actor and screenwriter (b. 1921)
 * 1992 – Michel Berger, French singer-songwriter and producer (b. 1947)
 * 1996 – Michel Debré, French lawyer and politician, 150th prime minister of France (b. 1912)
 * 1996  – Obdulio Varela, Uruguayan footballer and manager (b. 1917)
 * 1996  – Mohamed Farrah Aidid, Somalian general and politician, 5th president of Somalia (b. 1934)
 * 1997 – William S. Burroughs, American novelist, short story writer, and essayist (b. 1914)
 * 1997  – Harald Kihle, Norwegian painter and illustrator (b. 1905)
 * 1997  – Fela Kuti, Nigerian singer-songwriter and activist (b. 1938)
 * 1998 – Shari Lewis, American television host and puppeteer (b. 1933)
 * 1999 – Willie Morris, American writer (b. 1934)
 * 2003 – Peter Safar, Austrian-American physician and academic (b. 1924)
 * 2004 – Ferenc Berényi, Hungarian painter and academic (b. 1929)
 * 2004  – François Craenhals, Belgian illustrator (b. 1926)
 * 2004  – Heinrich Mark, Estonian lawyer and politician, 5th prime minister of Estonia in exile (b. 1911)
 * 2005 – Steven Vincent, American journalist and author (b. 1955)
 * 2007 – Chauncey Bailey, American journalist (b. 1950)
 * 2008 – Fujio Akatsuka, Japanese illustrator (b. 1935)
 * 2011 – José Sanchis Grau, Spanish author and illustrator (b. 1932)
 * 2012 – Gabriel Horn, English biologist and academic (b. 1927)
 * 2012  – Magnus Isacsson, Canadian director and producer (b. 1948)
 * 2012  – Jimmy Jones, American singer-songwriter (b. 1930)
 * 2012  – John Keegan, English historian and journalist (b. 1934)
 * 2012  – Bernd Meier, German footballer (b. 1972)
 * 2012  – Marguerite Piazza, American soprano (b. 1920)
 * 2013 – Julius L. Chambers, American lawyer and activist (b. 1936)
 * 2013  – Richard E. Dauch, American businessman, co-founded American Axle (b. 1942)
 * 2013  – Alla Kushnir, Russian–Israeli chess player (b. 1941)
 * 2014 – Ed Joyce, American journalist (b. 1932)
 * 2014  – Billie Letts, American author and educator (b. 1938)
 * 2014  – Barbara Prammer, Austrian social worker and politician (b. 1954)
 * 2014  – James Thompson, American-Finnish author (b. 1964)
 * 2015 – Forrest Bird, American pilot and engineer (b. 1921)
 * 2015  – Giovanni Conso, Italian jurist and politician, Italian Minister of Justice (b. 1922)
 * 2015  – Piet Fransen, Dutch footballer (b. 1936)
 * 2015  – Jack Spring, American baseball player (b. 1933)
 * 2016 – Terence Bayler, New Zealand actor (b. 1930)
 * 2016  – David Huddleston, American actor (b. 1930)
 * 2016  – Franciszek Macharski, Polish cardinal (b. 1927)
 * 2016  – Ahmed Zewail, Egyptian-American chemist and academic, Nobel Prize laureate (b. 1946)
 * 2017 – Judith Jones, American literary and cookbook editor (b. 1924)
 * 2020 – Suzanne Perlman, Hungarian-Dutch visual artist (b. 1922)
 * 2022 – Vin Scully, American sportscaster and game show host (b. 1927)
 * 2023 – Nitin Chandrakant Desai, Indian art director, production designer, and film and television producer (b. 1965)

Holidays and observances

 * Christian feast day:
 * Ahudemmeh (Syriac Orthodox Church).
 * Basil Fool for Christ (Russian Orthodox Church)
 * Justin Russolillo
 * Eusebius of Vercelli
 * Peter Faber
 * Peter Julian Eymard
 * Plegmund
 * Pope Stephen I
 * Portiuncola Indulgence ("Pardon of Assisi"), the plenary indulgence related to St. Francis of Assisi (Catholic Church).
 * Samuel David Ferguson (Episcopal Church)
 * August 2 (Eastern Orthodox liturgics)
 * Day of Azerbaijani cinema (Azerbaijan)
 * Our Lady of the Angels Day (Costa Rica)
 * Paratroopers Day (Russia)
 * Republic Day (North Macedonia)
 * Romani genocide-related observances, including:
 * Roma Holocaust Memorial Day (Council of Europe, European Parliament)

title: August 2


text:
Atlantic (disambiguation)

The Atlantic Ocean is the second largest of the world's oceans, that separates the old world from the new world.

Atlantic may also refer to:

In Canada

 * Atlantic, Nova Scotia
 * Atlantic Canada

In the United States

 * Atlantic, Iowa
 * Atlantic, Massachusetts
 * Atlantic, North Carolina, an unincorporated community in eastern Carteret County
 * Atlantic, Pennsylvania
 * Atlantic, Seattle, a neighborhood in Washington state
 * Atlantic, Virginia
 * Atlantic City, New Jersey
 * Atlantic County, New Jersey
 * Atlantic Peak (Colorado), a mountain

Companies and labels

 * Atlantic Books, an independent British publishing house
 * Atlantic Monthly Press, an American publishing house
 * Atlantic Entertainment Group, a defunct movie studio company
 * Atlantic FM, a former radio station serving Cornwall, United Kingdom
 * Atlantic Records, a record company

Music

 * The Atlantics, an Australian surf rock band formed in the early 1960s

Albums

 * Atlantic (Dufresne album)

Songs

 * "Atlantic" (song), by Keane
 * "Atlantic", a song by Björk from Vessel (DVD)
 * "Atlantic", a song by Thrice from Vheissu
 * "Atlantic", a song by Sleep Token from This Place Will Become Your Tomb

Other art, entertainment, and media

 * Atlantic (film), a 1929 black and white British film
 * The Atlantic, an American magazine founded as The Atlantic Monthly in 1857
 * Atlantic., a 2014 Dutch film
 * Atlantic (2015 film), an Irish documentary film, awarded Best Irish Documentary at the 2016 Dublin International Film Festival

Enterprises and organizations

 * Atlantic (cinema), a movie theater in Warsaw, Poland
 * Atlantic (toy company), a defunct Italian toy manufacturer
 * Atlantic (supermarkets), a defunct supermarket chain in Greece
 * Atlantic Broadband, a cable company in Massachusetts
 * Atlantic City Electric, a division of Elexon supplying electricity in New Jersey
 * Atlantic LNG, a liquefied natural gas producing company based in Trinidad and Tobago
 * Atlantic Petroleum, a former oil company in the United States
 * Atlantic Petroleum (Faroe Islands), an oil and gas production company
 * Atlantic Philanthropies, a defunct private foundation
 * Atlantic Superstore, a Canadian supermarket chain
 * Atlantic Technological University, north-western Ireland
 * Atlantic University, Virginia Beach, Virginia
 * Groupe Atlantic, a French climate control engineering company

Sports

 * Atlantic Championship Series, developmental open-wheel racing series in North America
 * Atlantic League of Professional Baseball, an American professional baseball league

Structures

 * Atlantic Building or Edificio Atlantic, a condominium building in Havana, Cuba
 * The Atlantic (Atlanta), a skyscraper in Atlanta, Georgia, United States

Airlines

 * Air Atlantic, a Canadian airline
 * Atlantic Airways, a Faroese airline company

Aircraft

 * Breguet Atlantic, a French long-range maritime patrol aircraft (1961)

Motor vehicles

 * Atlantic (1921 automobile), a defunct automobile company
 * Austin Atlantic, a British car produced by the Austin Motor Company from 1949 to 1952
 * Fisker Atlantic, a 2012 plug-in electric concept car

Railroads and trains

 * Atlantic (locomotive), name of an early steam-powered locomotive of the Baltimore and Ohio Railroad
 * Atlantic station (Los Angeles Metro)
 * Atlantic station (Staten Island Railway)
 * Atlantic (train), a named passenger train operated by Canadian Pacific Railway and later Via Rail
 * Atlantic, a type of steam locomotive with a 4-4-2 wheel arrangement (UIC classification 2B1)

Ships

 * Atlantic, any one of several vessels by that name
 * Atlantic (yacht), a three-masted gaff-rigged schooner
 * Atlantic 85-class lifeboats, lifeboats that serve the shores of the United Kingdom and Ireland as a part of the RNLI inshore fleet

Other uses

 * Atlantic (period) of palaeoclimatology
 * Atlantic languages (formerly West Atlantic), a language family in West Africa
 * Atlantic (horse), British-bred Thoroughbred racehorse of the 1870s

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Atlantic (disambiguation)


text:
Algebraic number

An algebraic number is a number that is a root of a non-zero polynomial (of finite degree) in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, $$(1 + \sqrt{5})/2$$, is an algebraic number, because it is a root of the polynomial $x2 − x − 1$. That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number $$1 + i$$ is algebraic because it is a root of $x4 + 4$.

All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as $\pi$ and $e$, are called transcendental numbers.

The set of algebraic numbers is countably infinite and has measure zero in the Lebesgue measure as a subset of the uncountable complex numbers. In that sense, almost all complex numbers are transcendental.

Examples

 * All rational numbers are algebraic. Any rational number, expressed as the quotient of an integer $a$ and a (non-zero) natural number $b$, satisfies the above definition, because $x = a⁄b$ is the root of a non-zero polynomial, namely $bx − a$.
 * Quadratic irrational numbers, irrational solutions of a quadratic polynomial $ax2 + bx + c$ with integer coefficients $a$, $b$, and $c$, are algebraic numbers. If the quadratic polynomial is monic ($a = 1$), the roots are further qualified as quadratic integers.
 * Gaussian integers, complex numbers $a + bi$ for which both $a$ and $b$ are integers, are also quadratic integers. This is because $a + bi$ and $a − bi$ are the two roots of the quadratic $x2 − 2ax + a2 + b2$.
 * A constructible number can be constructed from a given unit length using a straightedge and compass. It includes all quadratic irrational roots, all rational numbers, and all numbers that can be formed from these using the basic arithmetic operations and the extraction of square roots. (By designating cardinal directions for +1, −1, +$i$, and −$i$, complex numbers such as $$3+i \sqrt{2}$$ are considered constructible.)
 * Any expression formed from algebraic numbers using any combination of the basic arithmetic operations and extraction of $n$th roots gives another algebraic number.
 * Polynomial roots that cannot be expressed in terms of the basic arithmetic operations and extraction of $n$th roots (such as the roots of $x^{5} − x + 1$). That happens with many but not all polynomials of degree 5 or higher.
 * Values of trigonometric functions of rational multiples of π (except when undefined): for example, $cos π⁄7$, $cos 3

π⁄7$, and $cos 5

π⁄7$ satisfy $8x^{3} − 4x^{2} − 4x + 1 = 0$. This polynomial is irreducible over the rationals and so the three cosines are conjugate algebraic numbers. Likewise, $tan 3

π⁄16$, $tan 7

π⁄16$, $tan 11

π⁄16$, and $tan 15

π⁄16$ satisfy the irreducible polynomial $x^{4} − 4x^{3} − 6x^{2} + 4x + 1 = 0$, and so are conjugate algebraic integers. This is the equivalent of angles which, when measured in degrees, have rational numbers.
 * Some but not all irrational numbers are algebraic:
 * The numbers $$\sqrt{2}$$ and $$\frac{ \sqrt[3]{3} }{ 2 }$$ are algebraic since they are roots of polynomials $x^{2} − 2$ and $8x^{3} − 3$, respectively.
 * The golden ratio $φ$ is algebraic since it is a root of the polynomial $x^{2} − x − 1$.
 * The numbers π and e are not algebraic numbers (see the Lindemann–Weierstrass theorem).

Properties

 * If a polynomial with rational coefficients is multiplied through by the least common denominator, the resulting polynomial with integer coefficients has the same roots. This shows that an algebraic number can be equivalently defined as a root of a polynomial with either integer or rational coefficients.
 * Given an algebraic number, there is a unique monic polynomial with rational coefficients of least degree that has the number as a root. This polynomial is called its minimal polynomial. If its minimal polynomial has degree $n$, then the algebraic number is said to be of degree $n$. For example, all rational numbers have degree 1, and an algebraic number of degree 2 is a quadratic irrational.
 * The algebraic numbers are dense in the reals. This follows from the fact they contain the rational numbers, which are dense in the reals themselves.
 * The set of algebraic numbers is countable (enumerable), and therefore its Lebesgue measure as a subset of the complex numbers is 0 (essentially, the algebraic numbers take up no space in the complex numbers). That is to say, "almost all" real and complex numbers are transcendental.
 * All algebraic numbers are computable and therefore definable and arithmetical.
 * For real numbers $a$ and $b$, the complex number $a + bi$ is algebraic if and only if both $a$ and $b$ are algebraic.

Degree of simple extensions of the rationals as a criterion to algebraicity
For any $α$, the simple extension of the rationals by $α$, denoted by $$\Q(\alpha) \equiv \{\sum_{i=-{n_1}}^{n_2} \alpha^i q_i | q_i\in \Q, n_1,n_2\in \N\}$$, is of finite degree if and only if $α$ is an algebraic number.

The condition of finite degree means that there is a finite set $$\{a_i | 1\le i\le k\}$$ in $$\Q(\alpha)$$ such that $$\Q(\alpha) = \sum_{i=1}^k a_i \Q$$; that is, every member in $$\Q(\alpha)$$ can be written as $$\sum_{i=1}^k a_i q_i$$ for some rational numbers $$\{q_i | 1\le i\le k\}$$ (note that the set $$\{a_i\}$$ is fixed).

Indeed, since the $$a_i-s$$ are themselves members of $$\Q(\alpha)$$, each can be expressed as sums of products of rational numbers and powers of $α$, and therefore this condition is equivalent to the requirement that for some finite $$n$$, $$\Q(\alpha) = \{\sum_{i=-n}^n \alpha^{i} q_i | q_i\in \Q\}$$.

The latter condition is equivalent to $$\alpha^{n+1}$$, itself a member of $$\Q(\alpha)$$, being expressible as $$\sum_{i=-n}^n \alpha^i q_i$$ for some rationals $$\{q_i\}$$, so $$\alpha^{2n+1} = \sum_{i=0}^{2n} \alpha^i q_{i-n}$$ or, equivalently, $α$ is a root of $$x^{2n+1}-\sum_{i=0}^{2n} x^i q_{i-n}$$; that is, an algebraic number with a minimal polynomial of degree not larger than $$2n+1$$.

It can similarly be proven that for any finite set of algebraic numbers $$\alpha_1$$, $$\alpha_2$$... $$\alpha_n$$, the field extension $$\Q(\alpha_1, \alpha_2, ... \alpha_n)$$ has a finite degree.

Field
The sum, difference, product, and quotient (if the denominator is nonzero) of two algebraic numbers is again algebraic:

For any two algebraic numbers $α$, $β$, this follows directly from the fact that the simple extension $$\Q(\gamma)$$, for $$\gamma$$ being either $$\alpha+\beta$$, $$\alpha-\beta$$, $$\alpha\beta$$ or (for $$\beta\ne 0$$) $$\alpha/\beta$$, is a linear subspace of the finite-degree field extension $$\Q(\alpha,\beta)$$, and therefore has a finite degree itself, from which it follows (as shown above) that $$\gamma$$ is algebraic.

An alternative way of showing this is constructively, by using the resultant.

Algebraic numbers thus form a field $$\overline{\mathbb{Q}}$$ (sometimes denoted by $$\mathbb A$$, but that usually denotes the adele ring).

Algebraic closure
Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic. That can be rephrased by saying that the field of algebraic numbers is algebraically closed. In fact, it is the smallest algebraically-closed field containing the rationals and so it is called the algebraic closure of the rationals.

That the field of algebraic numbers is algebraically closed can be proven as follows: Let $β$ be a root of a polynomial $$ \alpha_0 + \alpha_1 x + \alpha_2 x^2 ... +\alpha_n x^n$$ with coefficients that are algebraic numbers $$\alpha_0$$, $$\alpha_1$$, $$\alpha_2$$... $$\alpha_n$$. The field extension $$\Q^\prime \equiv \Q(\alpha_1, \alpha_2, ... \alpha_n)$$ then has a finite degree with respect to $$\Q$$. The simple extension $$\Q^\prime(\beta)$$ then has a finite degree with respect to $$\Q^\prime$$ (since all powers of $β$ can be expressed by powers of up to $$\beta^{n-1}$$). Therefore, $$\Q^\prime(\beta) = \Q(\beta, \alpha_1, \alpha_2, ... \alpha_n)$$ also has a finite degree with respect to $$\Q$$. Since $$\Q(\beta)$$ is a linear subspace of $$\Q^\prime(\beta)$$, it must also have a finite degree with respect to $$\Q$$, so $β$ must be an algebraic number.

Numbers defined by radicals
Any number that can be obtained from the integers using a finite number of additions, subtractions, multiplications, divisions, and taking (possibly complex) $n$th roots where $n$ is a positive integer are algebraic. The converse, however, is not true: there are algebraic numbers that cannot be obtained in this manner. These numbers are roots of polynomials of degree 5 or higher, a result of Galois theory (see Quintic equations and the Abel–Ruffini theorem). For example, the equation:

 * $$x^5-x-1=0$$

has a unique real root that cannot be expressed in terms of only radicals and arithmetic operations.

Closed-form number
Algebraic numbers are all numbers that can be defined explicitly or implicitly in terms of polynomials, starting from the rational numbers. One may generalize this to "closed-form numbers", which may be defined in various ways. Most broadly, all numbers that can be defined explicitly or implicitly in terms of polynomials, exponentials, and logarithms are called "elementary numbers", and these include the algebraic numbers, plus some transcendental numbers. Most narrowly, one may consider numbers explicitly defined in terms of polynomials, exponentials, and logarithms – this does not include all algebraic numbers, but does include some simple transcendental numbers such as $e$ or ln 2.

Algebraic integers

An algebraic integer is an algebraic number that is a root of a polynomial with integer coefficients with leading coefficient 1 (a monic polynomial). Examples of algebraic integers are $$5 + 13 \sqrt{2},$$ $$2 - 6i,$$ and $\frac{1}{2}(1+i\sqrt{3}).$ Therefore, the algebraic integers constitute a proper superset of the integers, as the latter are the roots of monic polynomials $x − k$ for all $$k \in \mathbb{Z}$$. In this sense, algebraic integers are to algebraic numbers what integers are to rational numbers.

The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a ring. The name algebraic integer comes from the fact that the only rational numbers that are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers. If $K$ is a number field, its ring of integers is the subring of algebraic integers in $K$, and is frequently denoted as $O_{K}$. These are the prototypical examples of Dedekind domains.

Special classes

 * Algebraic solution
 * Gaussian integer
 * Eisenstein integer
 * Quadratic irrational number
 * Fundamental unit
 * Root of unity
 * Gaussian period
 * Pisot–Vijayaraghavan number
 * Salem number

title: Algebraic number


text:
Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.

Definition
In an algebraic structure such as a group, a ring, or vector space, an automorphism is simply a bijective homomorphism of an object into itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator.)

More generally, for an object in some category, an automorphism is a morphism of the object to itself that has an inverse morphism; that is, a morphism $$f: X\to X$$ is an automorphism if there is a morphism $$g: X\to X$$ such that $$g\circ f= f\circ g = \operatorname {id}_X,$$ where $$\operatorname {id}_X$$ is the identity morphism of $X$. For algebraic structures, the two definitions are equivalent; in this case, the identity morphism is simply the identity function, and is often called the trivial automorphism

Automorphism group
The automorphisms of an object $X$ form a group under composition of morphisms, which is called the automorphism group of $X$. This results straightforwardly from the definition of a category.

The automorphism group of an object $X$ in a category $C$ is often denoted $Aut_{C}(X)$$$, or simply Aut(X) if the category is clear from context.

Examples

 * In set theory, an arbitrary permutation of the elements of a set X is an automorphism. The automorphism group of X is also called the symmetric group on X.
 * In elementary arithmetic, the set of integers, Z, considered as a group under addition, has a unique nontrivial automorphism: negation. Considered as a ring, however, it has only the trivial automorphism. Generally speaking, negation is an automorphism of any abelian group, but not of a ring or field.
 * A group automorphism is a group isomorphism from a group to itself. Informally, it is a permutation of the group elements such that the structure remains unchanged. For every group G there is a natural group homomorphism G → Aut(G) whose image is the group Inn(G) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.
 * In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL(V). (The algebraic structure of all endomorphisms of V is itself an algebra over the same base field as V, whose invertible elements precisely consist of GL(V).)
 * A field automorphism is a bijective ring homomorphism from a field to itself.
 * The field $$\Q $$ of the rational numbers has no other automorphism than the identity, since an automorphism must fix the additive identity $0$ and the multiplicative identity $1$; the sum of a finite number of $1$ must be fixed, as well as the additive inverses of these sums (that is, the automorphism fixes all integers); finally, since every rational number is the quotient of two integers, all rational numbers must be fixed by any automorphism.
 * The field $$\R $$ of the real numbers has no other automorphism than the identity. Indeed, the rational numbers must be fixed by every automorphism, per above; an automorphism must preserve inequalities since $$x<y$$ is equivalent to $$\exists z\mid y-x=z^2,$$ and the latter property is preserved by every automorphism; finally every real number must be fixed since it is the least upper bound of a sequence of rational numbers.
 * The field $$\Complex$$ of the complex numbers, has a unique nontrivial automorphism, the that sends $$\R $$ into $$\R ,$$ the complex conjugation, but there are infinitely (uncountably) many "wild" automorphisms, if the axiom of choice is assumed.
 * The study of automorphisms of algebraic field extensions is the starting point and the main object of Galois theory.
 * The automorphism group of the quaternions (H) as a ring are the inner automorphisms, by the Skolem–Noether theorem: maps of the form a ↦ bab−1. This group is isomorphic to SO(3), the group of rotations in 3-dimensional space.
 * The automorphism group of the octonions (O) is the exceptional Lie group G2.
 * In graph theory an automorphism of a graph is a permutation of the nodes that preserves edges and non-edges. In particular, if two nodes are joined by an edge, so are their images under the permutation.
 * In geometry, an automorphism may be called a motion of the space. Specialized terminology is also used:
 * In metric geometry an automorphism is a self-isometry. The automorphism group is also called the isometry group.
 * In the category of Riemann surfaces, an automorphism is a biholomorphic map (also called a conformal map), from a surface to itself. For example, the automorphisms of the Riemann sphere are Möbius transformations.
 * An automorphism of a differentiable manifold M is a diffeomorphism from M to itself. The automorphism group is sometimes denoted Diff(M).
 * In topology, morphisms between topological spaces are called continuous maps, and an automorphism of a topological space is a homeomorphism of the space to itself, or self-homeomorphism (see homeomorphism group). In this example it is not sufficient for a morphism to be bijective to be an isomorphism.

History
One of the earliest group automorphisms (automorphism of a group, not simply a group of automorphisms of points) was given by the Irish mathematician William Rowan Hamilton in 1856, in his icosian calculus, where he discovered an order two automorphism, writing: "so that $\mu$ is a new fifth root of unity, connected with the former fifth root $\lambda$ by relations of perfect reciprocity."

Inner and outer automorphisms
In some categories—notably groups, rings, and Lie algebras—it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms.

In the case of groups, the inner automorphisms are the conjugations by the elements of the group itself. For each element a of a group G, conjugation by a is the operation φa : G → G given by φa(g) = aga−1 (or a−1ga; usage varies). One can easily check that conjugation by a is a group automorphism. The inner automorphisms form a normal subgroup of Aut(G), denoted by Inn(G); this is called Goursat's lemma.

The other automorphisms are called outer automorphisms. The quotient group Aut(G) / Inn(G) is usually denoted by Out(G); the non-trivial elements are the cosets that contain the outer automorphisms.

The same definition holds in any unital ring or algebra where a is any invertible element. For Lie algebras the definition is slightly different.

title: Automorphism


text:
Talk:Algebraic number

Subfields
Note our definition of subfield: "Let L be a field. A subfield of L is a subset K of L that is closed under the field operations of L and under taking inverses in L. In other words, K is a field with respect to the field operations inherited from L." The algebraic numbers are a subfield of the complex numbers; we need not verify the field axioms. — Arthur Rubin (talk) 16:49, 27 January 2016 (UTC)
 * The problem is that it is not mentioned that the complex numbers are a field, which makes the argument incomplete. Nxavar (talk) 08:40, 28 January 2016 (UTC)

Assessment comment (2006)
Substituted at 01:45, 5 May 2016 (UTC)

Real and Irrational
The are some disagreement between these two statements in wiki pages:

algebraic numbers: "A real algebraic number of degree 2 is a quadratic irrational."

quadratic irrational: "When c is positive, we get real quadratic irrational numbers, while a negative c gives complex quadratic irrational numbers which are not real numbers."

I suggest, to skip confusion, modify this page to "An algebraic number of degree 2 is a quadratic irrational." — Preceding unsigned comment added by 88.3.70.40 (talk) 09:29, 1 January 2017 (UTC)

Pi as an example
Recently someone put pi as an example of algebraic number, either because he/she did not understand the reference or a grief. Since it causes direct contradiction, I removed it for now. --Raxu360 (talk) 00:30, 30 June 2019 (UTC)

The malaise of Wikipedia mathematics articles.
An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently, by clearing denominators, with integer coefficients).

As with so many mathematical definitions in Wikipedia, this definition can only be understood by those who already know what the term means, and likely not all of them. (I have seen mathematical definitions in Wikipedia where I already knew what the term meant, but still couldn't understand the definition.)

The sole purpose of an encyclopedia is to convey information to those who do not already have it. This definition does not perform that function. Koro Neil (talk) 21:06, 8 October 2021 (UTC)

 * I would include that it is not the same as number representable in radicals. Valery Zapolodov (talk) 05:56, 25 December 2022 (UTC)

 * Do you have a proposed alternative? My suggestion would be to just get rid of all of the parentheticals as they make the sentence very awkward. Something like: “An algebraic number is a complex number that is the solution to a single-variable polynomial equation with rational coefficients. [... examples and further explanation unpacking some of those terms ...]” It looks like the definition has changed somewhat since this comment though. Did those changes solve your issue? –jacobolus (t) 00:43, 26 December 2022 (UTC)

Can we link to an online proof that they are enumerable
Says "The set of algebraic numbers is countable (enumerable)," with two old print refs. Can we link to an online proof ? - Rod57 (talk) 22:40, 19 September 2022 (UTC)

 * What online proof? Math. Stack. is not allowed, cause it is not a reliable source. Valery Zapolodov (talk) 05:54, 25 December 2022 (UTC)

title: Talk:Algebraic number


text:
Accordion

Accordions (from 19th-century German Akkordeon, from Akkord—"musical chord, concord of sounds") are a family of box-shaped musical instruments of the bellows-driven free reed aerophone type (producing sound as air flows past a reed in a frame). The essential characteristic of the accordion is to combine in one instrument a melody section, also called the diskant, usually on the right-hand keyboard, with an accompaniment or Basso continuo functionality on the left-hand. The musician normally plays the melody on buttons or keys on the right-hand side (referred to as the keyboard or sometimes the manual), and the accompaniment on bass or pre-set chord buttons on the left-hand side. A person who plays the accordion is called an accordionist.

The accordion belongs to the free-reed aerophone family. Other instruments in this family include the concertina, harmonica, and bandoneon. The concertina and bandoneon do not have the melody–accompaniment duality. The harmoneon is also related and, while having the descant vs. melody dualism, tries to make it less pronounced. The harmonium and American reed organ are in the same family, but are typically larger than an accordion and sit on a surface or the floor.

The accordion is played by compressing or expanding the bellows while pressing buttons or keys, causing pallets to open, which allow air to flow across strips of brass or steel, called reeds. These vibrate to produce sound inside the body. Valves on opposing reeds of each note are used to make the instrument's reeds sound louder without air leaking from each reed block.

The accordion is widely spread across the world because of the waves of migration from Europe to the Americas and other regions. In some countries (for example: Argentina, Brazil, Colombia, the Dominican Republic, Mexico, and Panama) it is used in popular music (for example: chamamé in Argentina; gaucho, forró, and sertanejo in Brazil; vallenato in Colombia; merengue in the Dominican Republic; and norteño in Mexico), whereas in other regions (such as Europe, North America, and other countries in South America) it tends to be more used for dance-pop and folk music.

In Europe and North America, some popular music acts also make use of the instrument. Additionally, the accordion is used in cajun, zydeco, jazz, and klezmer music, and in both solo and orchestral performances of classical music. Many conservatories in Europe have classical accordion departments. The oldest name for this group of instruments is harmonika, from the Greek harmonikos, meaning "harmonic, musical". Today, native versions of the name accordion are more common. These names refer to the type of accordion patented by Cyrill Demian, which concerned "automatically coupled chords on the bass side".

History
The accordion's basic form is believed to have been invented in Berlin, in 1822, by Christian Friedrich Ludwig Buschmann, although one instrument was discovered in 2006 that appears to have been built earlier.

The earliest history of the accordion in Russia is poorly documented. Nevertheless, according to Russian researchers, the earliest known simple accordions were made in Tula, Russia, by Ivan Sizov and Timofey Vorontsov around 1830, after they received an early accordion from Germany. By the late 1840s, the instrument was already very widespread; together the factories of the two masters were producing 10,000 instruments a year. By 1866, over 50,000 instruments were being produced yearly by Tula and neighbouring villages, and by 1874 the yearly production was over 700,000. By the 1860s, Novgorod, Vyatka and Saratov governorates also had significant accordion production. By the 1880s, the list included Oryol, Ryazan, Moscow, Tver, Vologda, Kostroma, Nizhny Novgorod and Simbirsk, and many of these places created their own varieties of the instrument.

The accordion is one of several European inventions of the early 19th century that use free reeds driven by a bellows. An instrument called accordion was first patented in 1829 by Cyrill Demian in Vienna. Demian's instrument bore little resemblance to modern instruments. It only had a left hand buttonboard, with the right hand simply operating the bellows. One key feature for which Demian sought the patent was the sounding of an entire chord by depressing one key. His instrument also could sound two different chords with the same key, one for each bellows direction (a bisonoric action). At that time in Vienna, mouth harmonicas with Kanzellen (chambers) had already been available for many years, along with bigger instruments driven by hand bellows. The diatonic key arrangement was also already in use on mouth-blown instruments. Demian's patent thus covered an accompanying instrument: an accordion played with the left hand, opposite to the way that contemporary chromatic hand harmonicas were played, small and light enough for travelers to take with them and used to accompany singing. The patent also described instruments with both bass and treble sections, although Demian preferred the bass-only instrument owing to its cost and weight advantages.

The accordion was introduced from Germany into Britain in about the year 1828. The instrument was noted in The Times in 1831 as one new to British audiences and was not favourably reviewed, but nevertheless it soon became popular. It had also become popular with New Yorkers by the mid-1840s.

After Demian's invention, other accordions appeared, some featuring only the right-handed keyboard for playing melodies. It took English inventor Charles Wheatstone to bring both chords and keyboard together in one squeezebox. His 1844 patent for what he called a concertina also featured the ability to easily tune the reeds from the outside with a simple tool.

The Austrian musician Adolf Müller described a great variety of instruments in his 1854 book Schule für Accordion. At the time, Vienna and London had a close musical relationship, with musicians often performing in both cities in the same year, so it is possible that Wheatstone was aware of this type of instrument and may have used them to put his key-arrangement ideas into practice.

Jeune's flutina resembles Wheatstone's concertina in internal construction and tone colour, but it appears to complement Demian's accordion functionally. The flutina is a one-sided bisonoric melody-only instrument whose keys are operated with the right hand while the bellows is operated with the left. When the two instruments are combined, the result is quite similar to diatonic button accordions still manufactured today.

Further innovations followed and continue to the present. Various buttonboard and keyboard systems have been developed, as well as voicings (the combination of multiple tones at different octaves), with mechanisms to switch between different voices during performance, and different methods of internal construction to improve tone, stability and durability. Modern accordions may incorporate electronics such as condenser microphones and tone and volume controls, so that the accordion can be plugged into a PA system or keyboard amplifier for live shows. Some 2010s-era accordions may incorporate MIDI sensors and circuitry, enabling the accordion to be plugged into a synth module and produce accordion sounds or other synthesized instrument sounds, such as piano or organ.

Construction
Accordions have many configurations and types. What may be easy to do with one type of accordion could be technically challenging or impossible with another, and proficiency with one layout may not translate to another.

The most obvious difference between accordions is their right-hand sides. Piano accordions use a piano-style musical keyboard; button accordions use a buttonboard. Button accordions are furthermore differentiated by their usage of a chromatic or diatonic buttonboard for the right-hand side.

Accordions may be either bisonoric, producing different pitches depending on the direction of bellows movement, or unisonoric, producing the same pitch in both directions. Piano accordions are unisonoric. Chromatic button accordions also tend to be unisonoric, while diatonic button accordions tend to be bisonoric, though notable exceptions exist.

Accordion size is not standardized, and may vary significantly from model to model. Accordions vary not only in their dimensions and weight, but also in number of buttons or keys present in the right- and left-hand keyboards. For example, piano accordions may have as few as 8 bass buttons (two rows of four), or up to 140 (seven rows of twenty) or beyond. Accordions also vary by their available registers and by their specific tuning and voicing.

Despite these differences, all accordions share a number of common components.

Bellows
[[File:Squeeze boxes accordion bandoneon concertina diatonic chromatic.jpg|thumb|265px|Bellows-driven instruments Piano accordions: 1, 2, 13

Diatonic button accordion: 3

Chromatic button accordions: 11, 12, 14

Digital accordions (V-Accordions, Roland Corporation): 11, 12, 13, 14

Bandoneon: 4

English concertina: 5

Anglo-German concertinas (Anglo concertinas): 6, 7, 8, 9, 10]] The bellows is the most recognizable part of the instrument, and the primary means of articulation. The production of sound in an accordion is in direct proportion to the motion of the bellows by the player. In a sense, the role of the bellows can be compared to the role of moving a violin's bow on bowed strings. For a more direct analogy, the bellows can be compared to the role of breathing for a singer. The bellows is located between the right- and left-hand keyboards, and is made from pleated layers of cloth and cardboard, with added leather and metal. It is used to create pressure and vacuum, driving air across the internal reeds and producing sound by their vibrations, applied pressure increases the volume.

The keyboard touch is not expressive and does not affect dynamics: all expression is effected through the bellows. Bellows effects include:
 * Volume control, including swells and fades
 * Repeated short, rapid changes of direction ("bellows shake"), which has been popularized by musicians such as Renato Borghetti (gaucho music) and Luiz Gonzaga, and extensively used in Forró, called resfulego in Brazil
 * Constant bellows motion while applying pressure at intervals
 * Constant bellows motion to produce clear tones with no resonance
 * Subtly changing the intonation to mimic the expressiveness of a singer
 * Using the bellows with the silent air button gives the sound of air moving ("whooshing"), which is sometimes used in contemporary compositions for this instrument

Body
The accordion's body consists of two boxes, commonly made of wood, joined by the bellows. These boxes house reed chambers for the right- and left-hand keyboards. Each side has grilles in order to facilitate the transmission of air in and out of the instrument, and to allow the sound to project. The grille at the right-hand side is usually larger and is often shaped for decorative purposes. The right-hand keyboard is normally used for playing the melody and the left-hand one for playing the accompaniment; however, skilled players can reverse these roles and play melodies with the left hand.

The size and weight of an accordion varies depending on its type, layout and playing range, which can be as small as to have only one or two rows of basses and a single octave on the right-hand keyboard, to the most common 120-bass accordion and through to large and heavy 160-bass free-bass converter models.

Pallet mechanism
The accordion is an aerophone. The keyboard mechanisms of the instrument either enable the air flow, or disable it:

Variable components
The term accordion covers a wide range of instruments, with varying components. All instruments have reed ranks of some format, apart from reedless digital accordions. Not all have switches to change registers or ranks, as some have only one treble register and one bass register. The most typical accordion is the piano accordion, which is used for many musical genres. Another type of accordion is the button accordion, which is used in musical traditions including Cajun, Conjunto and Tejano music, Swiss and Slovenian-Austro-German Alpine music, and Argentinian tango music. The Helikon-style accordion has multiple flared horns projecting out of the left side to strengthen the bass tone. The word "Helikon" refers to a deep-pitched tuba.

Right-hand keyboard systems
Different systems exist for the right-hand keyboard of an accordion, which is normally used for playing the melody (while it can also play chords). Some use a button layout arranged in one way or another, while others use a piano-style keyboard. Each system has different claimed benefits by those who prefer it. They are also used to define one accordion or another as a different "type":
 * Chromatic button accordions and the bayan, a Russian variant, use a buttonboard where notes are arranged chromatically. Two major systems exist, referred to as the B-system and the C-system (there are also regional variants). Rarely, some chromatic button accordions have a decorative right-hand keyboard in addition to the rows of buttons, an approach used by the virtuoso accordionist Pietro Frosini.
 * Diatonic button accordions use a buttonboard designed around the notes of diatonic scales in a small number of keys. The keys are often arranged in one row for each key available. Chromatic scales may be available by combining notes from different rows. The adjective "diatonic" is also commonly used to describe bisonic or bisonoric accordions—that is, instruments whose right-hand (and in some instances even bass) keys each sound two different notes depending on the direction of the bellows (for instance, producing major triad sequences while closing the bellows and dominant seventh or 7–9 while opening). Such is the case, for instance, with the Argentinian bandoneon, the Slovenian-Austro-German Steirische Harmonika, the Czech Heligonka Harmonika, the Italian organetto, the Swiss Schwyzerörgeli and the Anglo concertina.
 * Piano accordions use a musical keyboard similar to a piano, at right angles to the cabinet, the tops of the keys inward toward the bellows.
 * The rarely used bass accordion has only a right-hand keyboard, with ranks of 8', 16', and 32' reeds, with the lowest note being the deepest pitch on a pipe organ pedal keyboard (pedal C). It is intended for performing basslines in accordion orchestras.
 * The rarely used piccolo accordion also has only a right-hand keyboard.
 * 6-plus-6 accordions use a buttonboard with three rows of buttons in a "uniform" or "whole-tone" arrangement, generally known as a Jankó keyboard. The chromatic scale consists of two rows. The third row is a repetition of the first row, so there is the same fingering in all twelve scales. These accordions are produced only in special editions e.g. the logicordion produced by Harmona.

Left-hand keyboard systems
Different systems are also in use for the left-hand keyboard, which is normally used for playing the accompaniment. These usually use distinct bass buttons and often have buttons with concavities or studs to help the player navigate the layout despite not being able to see the buttons while playing. There are three general categories:
 * The Stradella bass system, also called standard bass, is arranged in a circle of fifths and uses single buttons for bass notes and additional rows of single buttons for preset major, minor, dominant seventh, and diminished chords. The dominant seventh and diminished chords are three-note chord voicings that omit the fifths of the chords.
 * The Belgian bass system is a variation used in Belgian chromatic accordions. It is also arranged in a circle of fifths but in reverse order. This system has three rows of basses, three rows of chord buttons allowing easier fingering for playing melodies, combined chords, better use of fingers one and five, and more space between the buttons. This system was rarely used outside of its native Belgium.
 * Various free-bass systems for greater access to playing melodies and complex basslines on the left-hand keyboard and to forming one's own chords note-by-note. These are often chosen for playing jazz and classical music. Some models can convert between free-bass and Stradella bass; this is called converter bass. The free-bass left hand notes are arranged chromatically in three rows with one additional duplicate row of buttons.
 * Luttbeg double-keyboard piano accordions have a piano keyboard layout on both the treble and bass sides. This allows pianists, most notably Duke Ellington, to double up on the accordion without difficulty. The Bercandeon is an improved version of that instrument, also making it a "keyboard bandoneon".
 * In 2021, a patent was published by Valerio Chiovarelli for a new bass system called the "Chiovarelli Jazz System". This system is a variation of the Stradella bass system where, instead of triads, the chordal buttons of this system produce bichords (chords with only 2 pitches instead of 3). The "Chiovarellia Jazz System" (or "CJS" for short) prioritizes the effectiveness of left hand accordion in jazz music, hence the name of the system, but according to the inventor, these chords can be useful when playing many varieties of music.

Reed ranks and switches

Inside the accordion are the reeds that generate the instrument tones. These are organized in different sounding banks, which can be further combined into registers producing differing timbres. All but the smaller accordions are equipped with switches that control which combination of reed banks operate, organized from high to low registers. Each register stop produces a separate sound timbre, many of which also differ in octaves or in how different octaves are combined. See the accordion reed ranks and switches article for further explanation and audio samples. All but the smaller accordions usually have treble switches. The larger and more expensive accordions often also have bass switches to give options for the reed bank on the bass side.

Classification of chromatic and piano type accordions
In describing or pricing an accordion, the first factor is size, expressed in number of keys on either side. For a piano type, this could for one example be 37/96, meaning 37 treble keys (three octaves plus one note) on the treble side and 96 bass keys. A second aspect of size is the width of the white keys, which means that even accordions with the same number of keys have keyboards of different lengths, ranging from 14 in for a child's accordion to 19 in for an adult-sized instrument. After size, the price and weight of an accordion is largely dependent on the number of reed ranks on either side, either on a cassotto or not, and to a lesser degree on the number of combinations available through register switches. The next, but important, factor is the quality of the reeds, the highest grade called "a mano" (meaning "hand-made"), the next "tipo a mano" ("like hand-made"), lower grades including "export" and several more.

Price is also affected by the use of costly woods, luxury decorations, and features such as a palm switch, grille mute, and so on. Some accordion makers sell a range of different models, from a less-expensive base model to a more costly luxury model. Typically, the register switches are described as Reeds: 5 + 3, meaning five reeds on the treble side and three on the bass, and Registers: 13 + M, 7, meaning 13 register buttons on the treble side plus a special "master" that activates all ranks, like the "tutti" or "full organ" switch on an organ, and seven register switches on the bass side. Another factor affecting the price is the presence of electronics, such as condenser microphones, volume and tone controls, or MIDI sensors and connections.

Straps
The larger piano and chromatic button accordions are usually heavier than other smaller squeezeboxes, and are equipped with two shoulder straps to make it easier to balance the weight and increase bellows control while sitting, and avoid dropping the instrument while standing. Other accordions, such as the diatonic button accordion, have only a single shoulder strap and a right hand thumb strap. All accordions have a (mostly adjustable) leather strap on the left-hand side to keep the player's hand in position while drawing the bellows. There are also straps above and below the bellows to keep it securely closed when the instrument is not being played.

Electronic and digital
In the 2010s, a range of electronic and digital accordions were introduced. They have an electronic sound module which creates the accordion sound, and most use MIDI systems to encode the keypresses and transmit them to the sound module. A digital accordion can have hundreds of sounds, which can include different types of accordions and even non-accordion sounds, such as pipe organ, piano, or guitar. Sensors are used on the buttons and keys, such as magnetic reed switches. Sensors are also used on the bellows to transmit the pushing and pulling of the bellows to the sound module. Digital accordions may have features not found in acoustic instruments, such as a piano-style sustain pedal, a modulation control for changing keys, and a portamento effect.

As an electronic instrument, these types of accordions are plugged into a PA system or keyboard amplifier to produce sound. Some digital accordions have a small internal speaker and amplifier, so they can be used without a PA system or keyboard amplifier, at least for practicing and small venues like coffeehouses. One benefit of electronic accordions is that they can be practiced with headphones, making them inaudible to other people nearby. On a digital accordion, the volume of the right-hand keyboard and the left-hand buttons can be independently adjusted.

Acoustic-digital hybrid accordions also exist. They are acoustic accordions (with reeds, bellows, and so on), but they also contain sensors, electronics, and MIDI connections, which provides a wider range of sound options. An acoustic-digital hybrid may be manufactured in this form, or it may be an acoustic accordion which has had aftermarket electronics sensors and connections added. Several companies sell aftermarket electronics kits, but they are typically installed by professional accordion technicians, because of the complex and delicate nature of the internal parts of an accordion.

Unusual accordions
Various hybrid accordions have been created between instruments of different buttonboards and actions. Many remain curiosities – only a few have remained in use:
 * The Schrammel accordion, used in Viennese chamber music and klezmer, which has the treble buttonboard of a chromatic button accordion and a bisonoric bass buttonboard, similar to an expanded diatonic button accordion
 * The Steirische Harmonika, a type of bisonoric diatonic button accordion particular to the Alpine folk music of Slovenia, Austria, the Czech Republic, the German state of Bavaria, and the Italian South Tyrol
 * The schwyzerörgeli or Swiss organ, which usually has a three-row diatonic treble and 18 unisonoric bass buttons in a bass/chord arrangement – a subset of the Stradella system in reverse order like the Belgian bass – that travel parallel to the bellows motion
 * The trikitixa of the Basque people, which has a two-row diatonic, bisonoric treble and a 12-button diatonic unisonoric bass
 * The British chromatic accordion, the favoured diatonic accordion in Scotland. While the right hand is bisonoric, the left hand follows the Stradella system. The elite form of this instrument is generally considered the German manufactured Shand Morino, produced by Hohner with the input of Sir Jimmy Shand
 * Pedal harmony, a type of accordion used sometimes in Polish folk music, which has a pair of pump organ-like bellows attached.
 * The Finnish composer and accordionist Veli Kujala developed a quarter tone accordion together with the Italian accordion manufacturer Pigini in 2005, and has written works for it. It deploys the same system as the concert accordion, with a scale of five octaves, each divided into 24 quarter tones. Other notable composers who have written concertos for the quarter tone accordion include Jukka Tiensuu and Sampo Haapamäki.

Manufacturing process
The most expensive accordions are typically fully hand-made, particularly the reeds; completely hand-made reeds have a better tonal quality than even the best automatically manufactured ones. Some accordions have been modified by individuals striving to bring a more pure sound out of low-end instruments, such as the ones improved by Yutaka Usui, a Japanese craftsman.

The manufacture of an accordion is only a partly automated process. In a sense, all accordions are handmade, since there is always some hand assembly of the small parts required. The general process involves making the individual parts, assembling the subsections, assembling the entire instrument, and final decorating and packaging.

Notable centres of production are the Italian cities of Stradella and Castelfidardo, with many small and medium size manufacturers especially at the latter. Castelfidardo honours the memory of Paolo Soprani who was one of the first large-scale producers. Maugein Freres has built accordions in the French town of Tulle since 1919, and the company is now the last complete-process manufacturer of accordions in France. German companies such as Hohner and Weltmeister made large numbers of accordions, but production diminished by the end of the 20th century. Hohner still manufactures its top-end models in Germany, and Weltmeister instruments are still handmade by HARMONA Akkordeon GmbH in Klingenthal.

Use in various music genres
The accordion has traditionally been used to perform folk or ethnic music, popular music, and transcriptions from the operatic and light-classical music repertoire. It was also used by the Kikuyu tribe in Kenya and is the main instrument in the traditional Mwomboko dance. Today the instrument is sometimes heard in contemporary pop styles, such as rock and pop-rock, and occasionally even in serious classical music concerts, as well as advertisements.

Use in traditional music

The accordion's popularity spread rapidly: it has mostly been associated with the common people, and was propagated by Europeans who emigrated around the world. The accordion in both button and piano forms became a favorite of folk musicians and has been integrated into traditional music styles all over the world: see the list of music styles that incorporate the accordion.

Notable jazz accordionists
Early jazz accordionists include Charles Melrose, who recorded Wailing Blues/Barrel House Stomp (1930, Voc. 1503) with the Cellar Boys; Buster Moten, who played second piano and accordion in the Bennie Moten orchestra; and Jack Cornell, who did recordings with Irving Mills. Later jazz accordionists from the United States include Steve Bach, Milton DeLugg, Orlando DiGirolamo, Angelo Di Pippo,   Dominic Frontiere, Guy Klucevsek, Yuri Lemeshev, Frank Marocco, Dr. William Schimmel, John Serry Sr., Lee Tomboulian, and Art Van Damme. French jazz accordionists include Richard Galliano, Bernard Lubat, and Vincent Peirani. Norwegian jazz accordionists include Asmund Bjørken, Stian Carstensen, Gabriel Fliflet, Frode Haltli, and Eivin One Pedersen.

Left hand techniques
The constraints of the Stradella bass system, limiting the left hand to preset chord buttons, is a barrier to some jazz chord conventions. Jazz accordionists expand the range of chord possibilities by using more than one chord button simultaneously, or by using combinations of a chord button and a bass note other than the typical root of the chord. An example of the former technique is used to play a minor seventh chord. To play an Am7(add9) chord, the Am and Em preset buttons are pressed simultaneously, along with an A bassnote. An example of the latter technique is used to play the half-diminished chord. To play an Eø7, a Gm preset button is pressed along with an E bassnote.

For the left hand, the free-bass system is used in jazz as a means of creating complex chord voicings. Jazz harmony that would otherwise be difficult to replicate with the Stradella bass system, such as tritone substitutions, become more accessible using a free-bass accordion.

Use in popular music
The accordion appeared in popular music from the 1900s to the 1960s. This half-century is often called the "golden age of the accordion". Five players, Pietro Frosini, the two brothers Count Guido Deiro and Pietro Deiro and Slovenian brothers Vilko Ovsenik and Slavko Avsenik, Charles Magnante were major influences at this time.

Most vaudeville theaters closed during the Great Depression, but accordionists during the 1930s–1950s taught and performed for radio. Included among this group was the concert virtuoso John Serry, Sr.  During the 1950s through the 1980s the accordion received significant exposure on television with performances by Myron Floren on The Lawrence Welk Show. In the late 1950s and early 1960s, the accordion declined in popularity because of the rise of rock and roll. The first accordionist to appear and perform at the Newport Jazz Festival was Angelo DiPippo. He can be seen playing his accordion in the motion picture The Godfather. He also composed and performed with his accordion on part of the soundtrack of Woody Allen's movie To Rome With Love. He was featured twice on The Tonight Show with Johnny Carson.

Richard Galliano is an internationally known accordionist whose repertoire covers jazz, tango nuevo, Latin, and classical. Some popular bands use the instrument to create distinctive sounds. A notable example is Grammy Award–winning parodist "Weird Al" Yankovic, who plays the accordion on many of his musical tracks, particularly his polkas. Yankovic was trained in the accordion as a child.

The accordion has also been used in the rock genre, most notably by John Linnell of They Might Be Giants, featuring more prominently in the band's earlier works. The instrument is still frequently used during live performances, and continues to make appearances in their studio albums. Accordion is also used in the music of the Dropkick Murphys and Gogol Bordello. Tom Waits used the Accordion extensively (Dr.William Schimmel) in "Raindogs" and "Frank's Wild Years", folk metal subgenre, and are otherwise generally rare. Full-time accordionists in folk metal seem even rarer, but they are still utilized for studio work, as flexible keyboardists are usually more accessible for live performances. The Finnish symphonic folk-metal band Turisas used to have a full-time accordionist, employing classical and polka sensibilities alongside a violinist. One of their accordionists, Netta Skog, is now a member of Ensiferum, another folk-metal band. Another Finnish metal band, Korpiklaani, invokes a type of Finnish polka called humppa, and also has a full-time accordionist. Sarah Kiener, the former hurdy-gurdy player for the Swiss melodic-death-folk metal band Eluveitie, played a Helvetic accordion known as a zugerörgeli.

Use in classical music
Although best known as a folk instrument, it has grown in popularity among classical composers. The earliest surviving concert piece is Thême varié très brillant pour accordéon methode Reisner, written in 1836 by Louise Reisner of Paris. Other composers, including the Russian Pyotr Ilyich Tchaikovsky, the Italian Umberto Giordano, and the American Charles Ives, wrote works for the diatonic button accordion.

The first composer to write specifically for the chromatic accordion was Paul Hindemith. In 1922, the Austrian Alban Berg included an accordion in Wozzeck, Op. 7. In 1937, the first accordion concerto was composed in Russia. Other notable composers have written for the accordion during the first half of the 20th century. Included among this group was the Italian-American John Serry Sr., whose Concerto for Free Bass Accordion was completed in 1964. In addition, the American accordionist Robert Davine composed his Divertimento for Flute, Clarinet, Bassoon and Accordion as a work for chamber orchestra. American composer William P. Perry featured the accordion in his orchestral suite Six Title Themes in Search of a Movie (2008). The experimental composer Howard Skempton began his musical career as an accordionist, and has written numerous solo works for it. In his work Drang (1999), British composer John Palmer pushed the expressive possibilities of the accordion/bayan. Luciano Berio wrote Sequenza XIII (1995) for accordionist Teodoro Anzellotti. Accordionists like Mogens Ellegaard, Joseph Macerollo, Nick Ariondo, Friedrich Lips, Hugo Noth, Dr. William Schimmel (also a composer), Stefan Hussong, Teodoro Anzellotti, and Geir Draugsvoll, encouraged composers to write new music for the accordion (solo and chamber music) and also started playing baroque music on the free bass accordion.

French composer Henri Dutilleux used an accordion in both his late song cycles Correspondences (2003) and Le Temps L'Horloge (2009). Russian-born composer Sofia Gubaidulina has composed solos, concertos, and chamber works for accordion. Astor Piazzolla's concert tangos are performed widely. Piazzolla performed on the bandoneon, but his works are also performed on or accordion. Dr. William schimmel and "The Tango Project" recorded a number of hit recordings and appeared in the movie Scent of a Woman with Al Pacino which earned Pacino an Oscar. Their recordings were used in many films.

Australia
The earliest mention of the novel accordion instrument in Australian music occurs in the 1830s. The accordion initially competed against cheaper and more convenient reed instruments such as mouth organ, concertina and melodeon. Frank Fracchia was an Australian accordion composer and copies of his works "My dear, can you come out tonight" and "Dancing with you" are preserved in Australian libraries. Other Australian composers who arranged music for accordion include Reginald Stoneham. The popularity of the accordion peaked in the late 1930s and continued until the 1950s. The accordion was particularly favoured by buskers.

Bosnia and Herzegovina
The accordion is a traditional instrument in Bosnia and Herzegovina. It is the dominant instrument used in sevdalinka, a traditional genre of folk music from Bosnia and Herzegovina.

Brazil
The accordion was brought to Brazil by settlers and immigrants from Europe, especially from Italy and Germany, who mainly settled in the south (Rio Grande do Sul, Santa Catarina and Paraná). The first instrument brought was a "Concertina" (a 120 button chromatic accordion). The instrument was popular in the 1950s, and it was common to find several accordions in the same house. There are many different configurations and tunes which were adapted from the cultures that came from Europe.

Accordion is the official symbol instrument of the Rio Grande do Sul state, where was voted by unanimity in the deputy chamber. During the boom of accordions there were around 65 factories in Brazil, where most of them (52) in the south, in Rio Grande do Sul state, with only 7 outside the south. One of the most famous and genuinely Brazilian brands was Accordeões Todeschini from Bento Gonçalves-RS, closed in 1973. The Todeschini accordion is very appreciated today and survives with very few maintainers. The most notable musicians of button accordions are Renato Borghetti, Adelar Bertussi, Albino Manique and Edson Dutra.

Compared to many other countries, the instrument is very popular in mainstream pop music. In some parts of the country, such as the northeast it is the most popular melodic instrument. As opposed to most European folk accordions, a very dry tuning is usually used in Brazil. Outside the south, the accordion (predominantly the piano accordion) is used in almost all styles of Forró (in particular in the subgenres of Xote and Baião) as the principal instrument, Luiz Gonzaga (the "King of the Baião") and Dominguinhos being among the notable musicians in this style from the northeast. In this musical style the typical combination is a trio of accordion, triangle and zabumba (a type of drum).

This style has gained popularity recently, in particular among the student population of the southeast of the country (in the Forró Universitário genre, with important exponents today being Falamansa, and trios such as Trio Dona Zefa, Trio Virgulino and Trio Alvorada). Moreover, the accordion is the principal instrument in Junina music (music of the São João Festival), with Mario Zan having been a very important exponent of this music. It is an important instrument in Sertanejo (and Caipira) music, which originated in the midwest and southeast of Brazil, and subsequently has gained popularity throughout the country.

China
The number of accordionists in China exceeds every other country in the world, and possibly every country combined. Introduced in 1926, the accordion has risen to popularity in China throughout the years, thanks to Russian teachers and its being a popular instrument in the People's Liberation Army, and remains popular.

In the late 20th century, the development of high performance standards for the accordion within China's halls of academe was also influenced by several American virtuosos including Robert Davine, who was invited by the Ministry of Culture of the People's Republic to present Master Classes and to broaden its national program of music for the accordion in 1984.

Colombia
The accordion is also a traditional instrument in Colombia, commonly associated with the vallenato and cumbia genres. The accordion has been used by tropipop musicians such as Carlos Vives, Andrés Cabas, Fonseca (singer) and Bacilos, as well as rock musicians such as Juanes and pop musicians as Shakira. Vallenato, who emerged in the early twentieth century in Valledupar, and have come to symbolize the folk music of Colombia.

Every year in April, Colombia holds one of the most important musical festivals in the country: the Vallenato Legend Festival. The festival holds contests for best accordion player. Once every decade, the "King of Kings" accordion competition takes place, where winners of the previous festivals compete for the highest possible award for a vallenato accordion player: the Pilonera Mayor prize. This is the world's largest competitive accordion festival.

Czech Republic
Accordion is often played at traditional Czech pubs, such as U Flekú, Prague.

Mexico
Norteño heavily relies on the accordion; it is a genre related to polka. Ramón Ayala, known in Mexico as the "King of the Accordion", is a norteño musician. Cumbia, which features the accordion, is also popular with musicians such as Celso Piña, creating a more contemporary style. U.S.-born Mexican musician Julieta Venegas incorporates the sound of the instrument into rock, pop and folk. She was influenced by her fellow Chicanos Los Lobos who also use the music of the accordion.

North Korea
According to Barbara Demick in Nothing to Envy, the accordion is known as "the people's instrument" and all North Korean teachers were expected to learn the accordion.

United States
Accordions are played in Tejano music, Cajun and Creole music, zydeco, klezmer, and polka.

During the post-World War II era from the 1940s to the 1960s, accordions were widely used in the United States for performances of traditional Western classical music within the configuration of large free-reed symphonic orchestras both in live performances on the concert hall stage and in phonograph recordings.

title: Accordion