View source for Al-Khwarizmi

{{Short description|Persian polymath (c. 780 – c. 850)}}
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{{Infobox academic
| image = Al Khorezmy.jpg
| image_size = 
| alt = 
| caption = Woodcut panel depicting al-Khwarizmi, 20th century
| name = Muḥammad ibn Mūsā al-Khwārizmī
| native_name = {{nobold|{{lang|fa|{{Script|Arab|محمد بن موسى خوارزمی}}|rtl=yes}}}}
| birth_date = {{circa|780}}
| death_date = {{circa|850|lk=no}}<ref>{{cite book |last1=Toomer |first1=Gerald J. |author1-link=Gerald J. Toomer |editor1-last=Gillispie |editor1-first=Charles Coulston |title=Dictionary of Scientific Biography |date=1970–1980 |isbn=978-0-684-16966-8|volume=VII |pages=358–365 |chapter=al-Khuwārizmī, Abu Ja'far Muḥammad ibn Mūsā|publisher=Scribner }}</ref><ref>{{cite book |last1=Vernet |first1=Juan |editor1-last=Gibb |editor1-first=H. A. R. |editor2-last=Kramers |editor2-first=J. H. |editor3-last=Lévi-Provençal |editor3-first=E. |editor4-last=Schacht |editor4-first=J. |title=The Encyclopaedia of Islam |date=1960–2005 |publisher=Brill |location=Leiden|volume=IV |pages=1070–1071 |edition=2nd |chapter=Al-Khwārizmī|oclc=399624}}</ref>
| era = [[Islamic Golden Age]]
| alma_mater = 
| school_tradition = 
| main_interests = {{hlist|[[Mathematics in the medieval Islamic world|Mathematics]]|[[Astronomy in the medieval Islamic world|astronomy]]|[[Geography and cartography in the medieval Islamic world|geography]]}}
| notable_ideas = Treatises on [[algebra]] and the [[Hindu–Arabic numeral system]]
| major_works = {{ubl|''[[Al-Jabr]]'' (820)|''[[Zij as-Sindhind]]'' (820)|''[[Geography (Ptolemy)|Kitab Surat al-Ard]]'' (833)}}
| influences = 
| influenced = [[Abu Kamil]] of Egypt,<ref name=MacTutor-AK>O'Connor, John J.; Robertson, Edmund F., [http://www-history.mcs.st-andrews.ac.uk/Biographies/Abu_Kamil.html "Abū Kāmil Shujā' ibn Aslam"] {{Webarchive|url=https://web.archive.org/web/20131211214159/http://www-history.mcs.st-andrews.ac.uk/Biographies/Abu_Kamil.html |date=11 December 2013 }}, MacTutor History of Mathematics archive, University of St Andrews.</ref> [[Nicolo Tartaglia]]
| awards = 
| occupation = [[House of Wisdom]] in [[Baghdad]] (appt.&nbsp;{{circa|820|lk=no}})
}}
{{Use dmy dates|date=March 2022}}
{{Use Oxford spelling|date=December 2023}}

'''Muhammad ibn Musa al-Khwarizmi'''{{refn|group=note|There is some confusion in the literature on whether al-Khwārizmī's full name is {{lang|fa|rtl=yes|ابو عبدالله محمد بن موسى خوارزمی}} {{transliteration|fa|ALA|Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī}} or {{lang|fa|rtl=yes|ابوجعفر محمد بن موسی خوارزمی}} {{transliteration|fa|ALA|Abū Ja'far Muḥammad ibn Mūsā al-Khwārizmī}}. [[Ibn Khaldun]] notes in his Prolegomena: "The first to write on this discipline [algebra] was Abu 'Abdallah al-Khuwarizmi. After him, there was Abu Kamil Shuja' b. Aslam. People followed in his steps."<ref>Ibn Khaldūn, [http://www.muslimphilosophy.com/ik/Muqaddimah/Table_of_Contents.htm The Muqaddimah: An introduction to history] {{Webarchive|url=https://web.archive.org/web/20160917023325/http://www.muslimphilosophy.com/ik/Muqaddimah/Table_of_Contents.htm |date=17 September 2016 }}, Translated from the Arabic by Franz Rosenthal, New York: Princeton (1958), Chapter VI:19.</ref> In the introduction to his critical commentary on Robert of Chester's Latin translation of al-Khwārizmī's ''Algebra'', L. C. Karpinski notes that Abū Ja'far Muḥammad ibn Mūsā refers to the eldest of the [[Banū Mūsā brothers]]. Karpinski notes in his review on (Ruska 1917) that in (Ruska 1918): "Ruska here inadvertently speaks of the author as Abū Ga'far M. b. M., instead of Abū Abdallah M. b. M." Donald Knuth writes it as {{transliteration|ar|ALA|Abū 'Abd Allāh Muḥammad ibn Mūsā al-Khwārizmī}} and quotes it as meaning "literally, 'Father of Abdullah, Mohammed, son of Moses, native of Khwārizm,'" citing previous work by Heinz Zemanek.<ref>{{cite book |first=Donald |last=Knuth |chapter=Basic Concepts |title=The Art of Computer Programming |volume=1 |edition=3rd |year=1997 |publisher=[[Addison-Wesley]] |isbn=978-0-201-89683-1 |page=1}}</ref>}} ({{langx|fa|محمد بن موسى خوارزمی}}; {{circa|lk=off|780|850}}), or simply '''al-Khwarizmi''', was a [[polymath]] who produced vastly influential Arabic-language works in [[Mathematics in the medieval Islamic world|mathematics]], [[Astronomy in the medieval Islamic world|astronomy]], and [[Geography and cartography in the medieval Islamic world|geography]]. Around 820, he worked at the [[House of Wisdom]] in [[Baghdad]], the contemporary capital city of the [[Abbasid Caliphate]].

His popularizing treatise on [[algebra]], compiled between 813–833 as ''[[Al-Jabr]]'' (''The Compendious Book on Calculation by Completion and Balancing''),<ref name=" Oaks">Oaks, J. (2009), "Polynomials and Equations in Arabic Algebra", ''Archive for History of Exact Sciences'', 63(2), 169–203.</ref>{{rp|171}} presented the first systematic solution of [[linear equation|linear]] and [[quadratic equation]]s. One of his achievements in [[algebra]] was his demonstration of how to solve quadratic equations by [[completing the square]], for which he provided geometric justifications.<ref name="Maher" />{{rp|14}} Because al-Khwarizmi was the first person to treat algebra as an independent discipline and introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation),<ref>(Boyer 1991, "The Arabic Hegemony" p. 229) "It is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation above. The word al-jabr presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word muqabalah is said to refer to "reduction" or "balancing"{{snd}}that is, the cancellation of like terms on opposite sides of the equation."</ref> he has been described as the father<ref name="Corbin 1998 44">{{Cite book|url=https://books.google.com/books?id=_VF0AgAAQBAJ&pg=PA44|title=The Voyage and the Messenger: Iran and Philosophy|last=Corbin|first=Henry|date=1998|publisher=North Atlantic|isbn=978-1-55643-269-9|page=44|access-date=19 October 2020|archive-date=28 March 2023|archive-url=https://web.archive.org/web/20230328222614/https://books.google.com/books?id=_VF0AgAAQBAJ&pg=PA44|url-status=live}}</ref><ref>[[Carl Benjamin Boyer|Boyer, Carl B.]], 1985. ''A History of Mathematics'', p.&nbsp;252. Princeton University Press. "Diophantus sometimes is called the father of algebra, but this title more appropriately belongs to al-Khowarizmi...", "...the Al-jabr comes closer to the elementary algebra of today than the works of either Diophantus or Brahmagupta..."</ref><ref>[[Solomon Gandz|Gandz, Solomon]], The sources of al-Khwarizmi's algebra, Osiris, i (1936), 263–277, "Al-Khwarizmi's algebra is regarded as the foundation and cornerstone of the sciences. In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers."</ref> or founder<ref>{{Cite journal|last=Katz|first=Victor J.|title=Stages in the History of Algebra with Implications for Teaching|url=https://eclass.uoa.gr/modules/document/file.php/MATH104/20010-11/HistoryOfAlgebra.pdf|journal=VICTOR J.KATZ, University of the District of Columbia Washington DC, USA|pages=190|via=University of the District of Columbia Washington DC, USA|quote=The first true algebra text which is still extant is the work on al-jabr and al-muqabala by Mohammad ibn Musa al-Khwarizmi, written in Baghdad around 825.|access-date=7 October 2017|archive-url=https://web.archive.org/web/20190327085930/https://eclass.uoa.gr/modules/document/file.php/MATH104/20010-11/HistoryOfAlgebra.pdf|archive-date=27 March 2019|url-status=dead}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=9HUDXkJIE3EC&pg=PA188|title=The Oxford History of Islam|last=Esposito|first=John L. |author-link=John Esposito |date=6 April 2000|publisher=[[Oxford University Press]]|isbn=978-0-19-988041-6|language=en|page=188|quote=Al-Khwarizmi is often considered the founder of algebra, and his name gave rise to the term algorithm.|access-date=29 September 2020|archive-date=28 March 2023|archive-url=https://web.archive.org/web/20230328222600/https://books.google.com/books?id=9HUDXkJIE3EC&pg=PA188|url-status=live}}</ref> of algebra. The English term ''algebra'' comes from the short-hand title of his aforementioned treatise ({{Lang|ar|الجبر|rtl=yes}} {{Transliteration|ar|Al-Jabr}},  {{Translation|"completion" or "rejoining"}}).<ref>{{Cite encyclopedia|last=Brentjes|first=Sonja|author-link=Sonja Brentjes|date=1 June 2007|title=Algebra|url=https://referenceworks.brillonline.com/entries/encyclopaedia-of-islam-3/algebra-COM_0030?s.num=11&s.f.s2_parent=s.f.book.encyclopaedia-of-islam-3&s.q=al+khwarazmi|encyclopedia=Encyclopaedia of Islam |edition=3rd|access-date=5 June 2019|archive-date=22 December 2019|archive-url=https://web.archive.org/web/20191222153702/https://referenceworks.brillonline.com/entries/encyclopaedia-of-islam-3/algebra-COM_0030?s.num=11&s.f.s2_parent=s.f.book.encyclopaedia-of-islam-3&s.q=al+khwarazmi|url-status=live}}</ref> His name gave rise to the English terms ''[[algorism]]'' and ''[[algorithm]]''; the Spanish, Italian, and Portuguese terms {{lang|und|algoritmo}}; and the Spanish term {{Lang|es|guarismo}}<ref>{{cite book|last=Knuth |first=Donald |url=http://historical.ncstrl.org/litesite-data/stan/CS-TR-80-786.pdf|title=Algorithms in Modern Mathematics and Computer Science|publisher=[[Springer-Verlag]]|date=1979|isbn=978-0-387-11157-5|author-link=Donald Knuth|url-status=dead|archive-url=https://web.archive.org/web/20061107213306/http://historical.ncstrl.org/litesite-data/stan/CS-TR-80-786.pdf|archive-date=7 November 2006}}</ref> and Portuguese term {{Lang|pt|algarismo}}, both meaning '[[numerical digit|digit]]'.<ref>{{Cite journal |last=Gandz |first=Solomon |author-link=Solomon Gandz |date=1926 |title=The Origin of the Term "Algebra" |journal=The American Mathematical Monthly |volume=33 |issue=9 |pages=437–440 |doi=10.2307/2299605 |jstor=2299605 |issn=0002-9890}}</ref>

In the 12th century, Latin translations of [[#Arithmetic|al-Khwarizmi's textbook on Indian arithmetic]] ({{Langx|la|Algorithmo de Numero Indorum|label=none}}), which codified the various [[Indian numerals]], introduced the [[decimal]]-based [[Positional notation|positional number system]] to the [[Western world]].<ref name="Struik 93">{{harvnb|Struik|1987| p= 93}}</ref> Likewise, ''Al-Jabr'', translated into Latin by the English scholar [[Robert of Chester]] in 1145, was used until the 16th century as the principal mathematical textbook of [[List of medieval universities|European universities]].<ref>{{Cite book|url=https://books.google.com/books?id=lQbcCwAAQBAJ|archive-url=https://web.archive.org/web/20191220170300/https://books.google.com/books?id=lQbcCwAAQBAJ|url-status=dead|archive-date=20 December 2019|title=History of the Arabs|last=Hitti |first=Philip Khuri |year=2002|isbn=978-1-137-03982-8|page=379| publisher=Palgrave Macmillan }}</ref><ref>{{Cite book|url=https://archive.org/details/isbn_9780781810159|url-access=registration|title=A History of the Islamic World|publisher=Hippocrene|last1=Hill |first1=Fred James |first2=Nicholas |last2=Awde|year=2003|isbn=978-0-7818-1015-9|page=[https://archive.org/details/isbn_9780781810159/page/55 55]|quote="The Compendious Book on Calculation by Completion and Balancing" (Hisab al-Jabr wa H-Muqabala) on the development of the subject cannot be underestimated. Translated into Latin during the twelfth century, it remained the principal mathematics textbook in European universities until the sixteenth century}}</ref><ref>{{Cite web|url=http://www.ms.uky.edu/~carl/ma330/project2/al-khwa21.html|title=Al-Khwarizmi |first1=Shawn |last1=Overbay |first2=Jimmy |last2=Schorer |first3=Heather |last3=Conger |publisher=[[University of Kentucky]]|archive-url=https://web.archive.org/web/20131212235239/http://www.ms.uky.edu/~carl/ma330/project2/al-khwa21.html|archive-date=12 December 2013|url-status=live}}</ref><ref>{{Cite web|url=http://www.sjsu.edu/people/patricia.backer/history/islam.htm|title=Islam Spain and the history of technology|website=|access-date=24 January 2018|archive-date=11 October 2018|archive-url=https://web.archive.org/web/20181011150650/http://www.sjsu.edu/people/patricia.backer/history/islam.htm|url-status=live}}</ref>

Al-Khwarizmi revised ''[[Geography (Ptolemy)|Geography]]'', the 2nd-century Greek-language treatise by the Roman polymath [[Ptolemy|Claudius Ptolemy]], listing the longitudes and latitudes of cities and localities.<ref>[[Bartel Leendert van der Waerden|van der Waerden, Bartel Leendert]] (1985). ''A History of Algebra: From al–Khwarizmi to Emmy Noether''. Berlin: Springer-Verlag.</ref>{{rp|9}} He further produced a set of astronomical tables and wrote about calendric works, as well as the [[astrolabe]] and the [[sundial]].<ref name="auto">{{harvnb|Arndt|1983|p=669}}</ref> Al-Khwarizmi made important contributions to [[trigonometry]], producing accurate [[sine and cosine]] tables and the first table of [[Tangent|tangents]].

== Life ==
[[Image:Madrid - Ciudad Universitaria, Monumento a Muhammad al-Juarismi.jpg|Monument to Muhammad ibn Musa al-Khwarizmi at Ciudad Universitaria of Madrid|thumb]]
Few details of al-Khwārizmī's life are known with certainty. [[Ibn al-Nadim]] gives his birthplace as [[Khwarazm]], and he is generally thought to have come from this region.<ref name="Science and medicine" /><ref>{{cite book |last1=Oaks |first1=Jeffrey A. |editor1-last=Kalin |editor1-first=Ibrahim |title=The Oxford Encyclopedia of Philosophy, Science, and Technology in Islam |date=2014 |publisher=[[Oxford University Press]] |location=[[Oxford]] |isbn=978-0-19-981257-8 |volume=1 |pages=451–459 |chapter=Khwārizmī |chapter-url-access=registration |chapter-url=https://www.academia.edu/27227712 |access-date=6 September 2021 |archive-date=30 January 2022 |archive-url=https://web.archive.org/web/20220130123536/https://www.academia.edu/27227712 |url-status=live }}<br />"''Ibn al-Nadīm and Ibn al-Qifṭī relate that al-Khwārizmī's family came from Khwārizm, the region south of the Aral sea''."<br /> Also → al-Nadīm, Abu'l-Faraj (1871–1872). ''Kitāb al-Fihrist'', ed. Gustav Flügel, Leipzig: Vogel, p. [https://archive.org/details/KitabAlFihrist/page/n447/mode/2up 274]. al-Qifṭī, Jamāl al-Dīn (1903). ''Taʾrīkh al-Hukamā'', eds. August Müller & Julius Lippert, Leipzig: Theodor Weicher, p. [https://archive.org/details/TarikhAlHukama/page/n237/mode/2up 286].</ref><ref name="Dodge">{{citation| editor-last=[[Bayard Dodge|Dodge]] | editor-first=Bayard| translator-last=Dodge |title=The Fihrist of al-Nadīm: A Tenth-Century Survey of Islamic Culture | publisher=[[Columbia University Press]] | place=New York | year=1970 |volume=2 }}</ref> Of [[Persians|Persian]] stock,<ref>{{cite book|author=Clifford A. Pickover|title=The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics|url=https://books.google.com/books?id=JrslMKTgSZwC&pg=PA84|year=2009|publisher=[[Sterling Publishing Company, Inc.]]|isbn=978-1-4027-5796-9|page=84|access-date=19 October 2020|archive-date=28 March 2023|archive-url=https://web.archive.org/web/20230328222600/https://books.google.com/books?id=JrslMKTgSZwC&pg=PA84|url-status=live}}</ref><ref name="Science and medicine">{{cite journal|last1=Saliba|first1=George|title=Science and medicine|journal=Iranian Studies|date=September 1998|volume=31|issue=3–4|pages=681–690|doi=10.1080/00210869808701940|quote=Take, for example, someone like Muhammad b. Musa al-Khwarizmi (fl. 850) may present a problem for the EIr, for although he was obviously of Persian descent, he lived and worked in Baghdad and was not known to have produced a single scientific work in Persian.}}</ref><ref>A History of Science in World Cultures: Voices of Knowledge. Routledge. Page 228. "Mohammed ibn Musa al-Khwarizmi (780–850) was a Persian astronomer and mathematician from the district of Khwarism (Uzbekistan area of Central Asia)."</ref><ref>{{cite book|last1=Ben-Menahem|first1=Ari|author-link1=Ari Ben-Menahem|title=Historical Encyclopedia of Natural and Mathematical Sciences|date=2009|publisher=Springer|location=Berlin|isbn=978-3-540-68831-0|pages=942–943|edition=1st|quote=Persian mathematician Al-Khowarizmi}}</ref><ref>{{cite book |last1=Wiesner-Hanks |first1=Merry E. |last2=Ebrey |first2=Patricia Buckley |last3=Beck |first3=Roger B. |last4=Davila |first4=Jerry |last5=Crowston |first5=Clare Haru |last6=McKay |first6=John P. |author1-link=Merry Wiesner-Hanks |author2-link=Patricia Buckley Ebrey |author6-link=John P. McKay |title=A History of World Societies |date=2017 |publisher=Bedford/St. Martin's |page=419 |edition=11th |quote=Near the beginning of this period the Persian scholar al-Khwarizmi (d. ca. 850) harmonized Greek and Indian findings to produce astronomical tables that formed the basis for later Eastern and Western research.}}</ref> his name means 'from Khwarazm', a region that was part of [[Greater Iran]],<ref>Encycloaedia Iranica-online, s.v. "[https://iranicaonline.org/articles/chorasmia-ii CHORASMIA, ii. In Islamic times] {{Webarchive|url=https://web.archive.org/web/20210902091627/https://iranicaonline.org/articles/chorasmia-ii |date=2 September 2021 }}," by [[Clifford Edmund Bosworth|Clifford E. Bosworth]].</ref> and is now part of [[Turkmenistan]] and [[Uzbekistan]].<ref>{{cite book |last1=Bosworth |first1=Clifford Edmund |author1-link=Clifford Edmund Bosworth |editor1-last=Gibb |editor1-first=H. A. R. |editor2-last=Kramers |editor2-first=J. H. |editor3-last=Lévi-Provençal |editor3-first=E. |editor4-last=Schacht |editor4-first=J. |title=The Encyclopaedia of Islam |date=1960–2005 |publisher=Brill |location=Leiden|volume=IV |pages=1060–1065 |edition=2nd |chapter=Khwārazm|oclc=399624}}</ref>

[[Al-Tabari]] gives his name as Muḥammad ibn Musá al-Khwārizmī al-[[Majus|Majūsī]] al-Quṭrubbullī ({{lang|ar|محمد بن موسى الخوارزميّ المجوسـيّ القطربّـليّ}}). The [[epithet]] ''al-Qutrubbulli'' could indicate he might instead have come from Qutrubbul (Qatrabbul),<ref>"Iraq After the Muslim Conquest", by [[Michael G. Morony]], {{isbn|1-59333-315-3}} (a 2005 facsimile from the original 1984 book), [https://books.google.com/books?id=uhjSiRAwGuEC&dq=qatrabbul&pg=PA145 p.&nbsp;145] {{Webarchive|url=https://web.archive.org/web/20140627081909/http://books.google.com/books?id=uhjSiRAwGuEC&pg=PA145&dq=qatrabbul |date=27 June 2014 }}</ref> near Baghdad. However, [[Roshdi Rashed]] denies this:<ref>{{Cite book|last=Rashed|first=Roshdi |author-link=Roshdi Rashed|url=https://books.google.com/books?id=JXbXRKRY_uAC|title=Arab Civilization: Challenges and Responses : Studies in Honor of Constantine K. Zurayk|date=1988|publisher=SUNY Press|isbn=978-0-88706-698-6|editor-last=Zurayq|editor-first=Qusṭanṭīn|page=108|contribution=al-Khwārizmī's Concept of Algebra|editor2-last=Atiyeh|editor2-first=George Nicholas|editor3-last=Oweiss|editor3-first=Ibrahim M.|contribution-url=https://books.google.com/books?id=JXbXRKRY_uAC&pg=PA108|access-date=19 October 2015|archive-date=28 March 2023|archive-url=https://web.archive.org/web/20230328222551/https://books.google.com/books?id=JXbXRKRY_uAC|url-status=live}}</ref>
{{blockquote|There is no need to be an expert on the period or a philologist to see that al-Tabari's second citation should read "Muhammad ibn Mūsa al-Khwārizmī ''and'' al-Majūsi al-Qutrubbulli," and that there are two people (al-Khwārizmī and al-Majūsi al-Qutrubbulli) between whom the letter ''wa'' [Arabic '{{lang|ar|و}}' for the conjunction '[[wikt:و#Etymology 2|and]]'] has been omitted in an early copy. This would not be worth mentioning if a series of errors concerning the personality of al-Khwārizmī, occasionally even the origins of his knowledge, had not been made. Recently, [[Gerald J. Toomer|G.J. Toomer]] ... with naive confidence constructed an entire fantasy on the error which cannot be denied the merit of amusing the reader.}}

On the other hand, [[David A. King (historian)|David A. King]] affirms his [[Nisba (onomastics)|nisba]] to Qutrubul, noting that he was called al-Khwārizmī al-Qutrubbulli because he was born just outside of Baghdad.<ref>{{Cite AV media| people = [[David A. King (historian)|King, David A.]] | title = Astronomy in the Service of Islam| date = 7 March 2018| time = 20:51| publisher = Al-Furqān Islamic Heritage Foundation – Centre for the Study of Islamic Manuscripts| url = https://al-furqan.com/events/astronomy-in-the-service-of-islam/| quote = I mention another name of Khwarizmi to show that he didn't come from Central Asia. He came from Qutrubul, just outside Baghdad. He was born there, otherwise he wouldn't be called al-Qutrubulli. Many people say he came from Khwarazm, tsk-tsk.| access-date = 26 November 2021| archive-date = 1 December 2021| archive-url = https://web.archive.org/web/20211201005353/https://al-furqan.com/events/astronomy-in-the-service-of-islam/| url-status = live}}</ref>

Regarding al-Khwārizmī's religion, Toomer writes:<ref name="toomer">{{harvnb|Toomer|1990}}</ref>

{{blockquote|Another epithet given to him by al-Ṭabarī, "al-Majūsī," would seem to indicate that he was an adherent of the old [[Zoroastrianism|Zoroastrian religion]]. This would still have been possible at that time for a man of Iranian origin, but the pious preface to al-Khwārizmī's ''Algebra'' shows that he was an orthodox [[Muslim]], so al-Ṭabarī's epithet could mean no more than that his forebears, and perhaps he in his youth, had been Zoroastrians.|author=|title=|source=}}

[[Ibn al-Nadim|Ibn al-Nadīm]]'s {{Lang|ar-latn|[[Al-Fihrist]]}} includes a short biography on al-Khwārizmī together with a list of his books. Al-Khwārizmī accomplished most of his work between 813 and 833. After the [[Muslim conquest of Persia]], Baghdad had become the centre of scientific studies and trade. Around 820 CE, he was appointed as the astronomer and head of the library of the [[House of Wisdom]].<ref name="Maher">Maher, P. (1998), "From Al-Jabr to Algebra", ''Mathematics in School'', 27(4), 14–15.</ref>{{rp|14}} The House of Wisdom was established by the [[Abbasid Caliphate|Abbasid]] [[Al-Ma'mun|Caliph al-Ma'mūn]]. Al-Khwārizmī studied sciences and mathematics, including the translation of [[Greek language|Greek]] and [[Sanskrit]] scientific manuscripts. He was also a historian who is cited by the likes of [[al-Tabari]] and [[Ibn Abi Tahir]].<ref>{{The History of al-Tabari | volume = 32 | page = 158}}</ref>

During the reign of [[al-Wathiq]], he is said to have been involved in the first of two embassies to the [[Khazars]].<ref>{{Cite book| publisher = BRILL| isbn = 978-90-474-2145-0| last1 = Golden| first1 = Peter| last2 = Ben-Shammai| first2 = Haggai| last3 = Roná-Tas| first3 = András| title = The World of the Khazars: New Perspectives. Selected Papers from the Jerusalem 1999 International Khazar Colloquium| date = 13 August 2007|page=376}}</ref> [[Douglas Morton Dunlop]] suggests that Muḥammad ibn Mūsā al-Khwārizmī might have been the same person as Muḥammad ibn Mūsā ibn Shākir, the eldest of the three [[Banū Mūsā brothers]].<ref>{{harvnb|Dunlop|1943}}</ref>

== Contributions ==
[[File:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|thumb|A page from al-Khwārizmī's ''Algebra'']]

Al-Khwārizmī's contributions to mathematics, geography, astronomy, and [[cartography]] established the basis for innovation in algebra and [[trigonometry]]. His systematic approach to solving linear and quadratic equations led to ''algebra'', a word derived from the title of his book on the subject, ''Al-Jabr''.<ref>{{Cite web |url=http://sharif.edu/~tabesh/math-education.pdf |title=Mathematics Education in Iran From Ancient to Modern |author=Yahya Tabesh |author2=Shima Salehi |website=Sharif University of Technology |access-date=16 April 2018 |archive-date=16 April 2018 |archive-url=https://web.archive.org/web/20180416200423/http://sharif.edu/~tabesh/math-education.pdf |url-status=live }}</ref>

''On the Calculation with Hindu Numerals,'' written about 820, was principally responsible for spreading the [[Hindu–Arabic numeral system]] throughout the Middle East and Europe. When the work was translated into Latin in the 12th century as ''Algoritmi de numero Indorum'' (Al-Khwarizmi on the Hindu art of reckoning), the term "algorithm" was introduced to the Western world.<ref>{{Cite book |last=Presner |first=Todd |url=https://books.google.com/books?id=f9X9EAAAQBAJ |title=Ethics of the Algorithm: Digital Humanities and Holocaust Memory |date=2024-09-24 |publisher=Princeton University Press |isbn=978-0-691-25896-6 |pages=20 |language=en}}</ref><ref>{{harvnb|Daffa|1977}}</ref><ref name=":0">{{Cite book|last=Clegg|first=Brian|url=https://books.google.com/books?id=by-4DwAAQBAJ&pg=PA61|title=Scientifica Historica: How the world's great science books chart the history of knowledge|date=1 October 2019|publisher=Ivy Press|isbn=978-1-78240-879-6|pages=61|language=en|access-date=30 December 2021|archive-date=28 March 2023|archive-url=https://web.archive.org/web/20230328222610/https://books.google.com/books?id=by-4DwAAQBAJ&pg=PA61|url-status=live}}</ref>

Some of his work was based on Persian and [[Babylonia]]n astronomy, [[Indian numbering system|Indian numbers]], and [[Greek mathematics]].

Al-Khwārizmī systematized and corrected [[Ptolemy]]'s data for Africa and the Middle East. Another major book was ''Kitab surat al-ard'' ("The Image of the Earth"; translated as Geography), presenting the coordinates of places based on those in the [[Geography (Ptolemy)|''Geography'' of Ptolemy]], but with improved values for the [[Mediterranean Sea]], Asia, and Africa.<ref>{{Cite web|url=https://www.worldhistoryedu.com/al-khwarizmi-biography-notable-achievements-facts/|title=Al-Khwārizmī - Biography, Notable Achievements & Facts|first=World History|last=Edu|date=28 September 2022}}</ref>

He wrote on mechanical devices like the <!-- [[clock]], al-Biruni? -->[[astrolabe]]<ref>Joseph Frank, ''al-Khwarizmi über das Astrolab'', 1922.</ref> and [[sundial]].<ref name="auto"/><!-- His other contributions include tables of [[trigonometric function]]s, refinements in the geometric representation of [[conic sections]], and aspects of the [[calculus of two errors]]. More al-Biruni? al-Khwārizmī's made a table of sien values. --> He assisted a project to determine the circumference of the Earth and in making a world map for [[al-Ma'mun]], the caliph, overseeing 70 geographers.<ref>{{cite encyclopedia|access-date=30 May 2008|url=https://www.britannica.com/eb/article-9045366|title=al-Khwarizmi|encyclopedia=[[Encyclopædia Britannica]]|archive-date=5 January 2008|archive-url=https://web.archive.org/web/20080105123350/http://www.britannica.com/eb/article-9045366|url-status=live}}</ref> When, in the 12th century, his works spread to Europe through Latin translations, it had a profound impact on the advance of mathematics in Europe.<ref>{{Cite web|url=https://www.britannica.com/biography/al-Khwarizmi|title=Al-Khwarizmi &#124; Biography & Facts &#124; Britannica|date=1 December 2023|website=www.britannica.com}}</ref>

=== Algebra ===
{{Main|Al-Jabr}}
{{Further|Latin translations of the 12th century|Mathematics in medieval Islam|Science in the medieval Islamic world}}
{{multiple image
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| footer = Left: The original Arabic print manuscript of the ''Book of Algebra'' by Al-Khwārizmī. Right: A page from ''The Algebra of Al-Khwarizmi'' by Fredrick Rosen, in English.
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''Al-Jabr (The Compendious Book on Calculation by Completion and Balancing'', {{langx|ar|الكتاب المختصر في حساب الجبر والمقابلة}} {{transliteration|ar|ALA|al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala}}) is a mathematical book written approximately 820 CE. It was written with the encouragement of [[Al-Ma'mun|Caliph al-Ma'mun]] as a popular work on calculation and is replete with examples and applications to a range of problems in trade, surveying and legal inheritance.<ref name=Algebra_1831_translation_rosen>{{cite web
 |url=http://www.wilbourhall.org/index.html#algebra
 |work=1831 English Translation
 |title=The Compendious Book on Calculation by Completion and Balancing, al-Khwārizmī
 |first=Frederic
 |last=Rosen
 |access-date=14 September 2009
 |archive-date=16 July 2011
 |archive-url=https://web.archive.org/web/20110716101515/http://www.wilbourhall.org/index.html#algebra
 |url-status=live
 }}</ref> The term "algebra" is derived from the name of one of the basic operations with equations ({{transliteration|ar|ALA|al-jabr}}, meaning "restoration", referring to adding a number to both sides of the equation to consolidate or cancel terms) described in this book. The book was translated in Latin as ''Liber algebrae et almucabala'' by [[Robert of Chester]] ([[Segovia]], 1145) hence "algebra", and by [[Gerard of Cremona]]. A unique Arabic copy is kept at Oxford and was translated in 1831 by F. Rosen. A Latin translation is kept in Cambridge.<ref>{{cite journal|author=Karpinski, L.C.|date=1912|title=History of Mathematics in the Recent Edition of the Encyclopædia Britannica|journal=Science|volume=35|issue=888|pages=29–31|author-link=L. C. Karpinski|bibcode=1912Sci....35...29K|doi=10.1126/science.35.888.29|pmid=17752897|url=https://zenodo.org/record/1448076|access-date=29 September 2020|archive-date=30 October 2020|archive-url=https://web.archive.org/web/20201030113041/https://zenodo.org/record/1448076|url-status=live}}</ref>

It provided an exhaustive account of solving polynomial equations up to the second degree,{{sfn|Boyer|1991|p=[https://archive.org/details/historyofmathema00boye/page/228 228]|ps=: "The Arabs in general loved a good clear argument from premise to conclusion, as well as systematic organization — respects in which neither Diophantus nor the Hindus excelled."}} and discussed the fundamental method of "reduction" and "balancing", referring to the transposition of terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation.<ref name=Boyer-229>{{Harv|Boyer|1991|loc="The Arabic Hegemony" p.&nbsp;229}} "It is not certain just what the terms ''al-jabr'' and ''muqabalah'' mean, but the usual interpretation is similar to that implied in the translation above. The word ''al-jabr'' presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word ''muqabalah'' is said to refer to "reduction" or "balancing" — that is, the cancellation of like terms on opposite sides of the equation."</ref>

Al-Khwārizmī's method of solving linear and quadratic equations worked by first reducing the equation to one of six standard forms (where ''b'' and ''c'' are positive integers)
* squares equal roots (''ax''<sup>2</sup> = ''bx'')
* squares equal number (''ax''<sup>2</sup> = ''c'')
* roots equal number (''bx'' = ''c'')
* squares and roots equal number (''ax''<sup>2</sup> + ''bx'' = ''c'')
* squares and number equal roots (''ax''<sup>2</sup> + ''c'' = ''bx'')
* roots and number equal squares (''bx'' + ''c'' = ''ax''<sup>2</sup>)

by dividing out the coefficient of the square and using the two operations {{transliteration|ar|ALA|al-jabr}} ({{langx|ar|الجبر}} "restoring" or "completion") and {{transliteration|ar|ALA|al-muqābala}} ("balancing"). {{transliteration|ar|ALA|Al-jabr}} is the process of removing negative units, roots and squares from the equation by adding the same quantity to each side. For example, ''x''<sup>2</sup> = 40''x''&nbsp;−&nbsp;4''x''<sup>2</sup> is reduced to 5''x''<sup>2</sup> = 40''x''. {{transliteration|ar|ALA|Al-muqābala}} is the process of bringing quantities of the same type to the same side of the equation. For example, ''x''<sup>2</sup>&nbsp;+&nbsp;14 = ''x''&nbsp;+&nbsp;5 is reduced to ''x''<sup>2</sup>&nbsp;+&nbsp;9 = ''x''.

The above discussion uses modern mathematical notation for the types of problems that the book discusses. However, in al-Khwārizmī's day, most of this notation [[History of mathematical notation|had not yet been invented]], so he had to use ordinary text to present problems and their solutions. For
example, for one problem he writes, (from an 1831 translation)
{{blockquote|If some one says: "You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times." Computation: You say, ten less a thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts.<ref name=Algebra_1831_translation_rosen />}}
In modern notation this process, with ''x'' the "thing" ({{lang|ar|شيء}} ''shayʾ'') or "root", is given by the steps,
:<math>(10-x)^2=81 x</math>
:<math>100 + x^2 - 20 x = 81 x</math>
:<math>x^2+100=101 x</math>

Let the roots of the equation be ''x'' = ''p'' and ''x = q''. Then <math>\tfrac{p+q}{2}=50\tfrac{1}{2}</math>, <math>pq =100</math> and
:<math>\frac{p-q}{2} = \sqrt{\left(\frac{p+q}{2}\right)^2 - pq}=\sqrt{2550\tfrac{1}{4} - 100}=49\tfrac{1}{2}</math>
So a root is given by
:<math>x=50\tfrac{1}{2}-49\tfrac{1}{2}=1</math>

Several authors have published texts under the name of {{transliteration|ar|ALA|Kitāb al-jabr wal-muqābala}}, including [[Abū Ḥanīfa Dīnawarī]], [[Abū Kāmil]], Abū Muḥammad al-'Adlī, Abū Yūsuf al-Miṣṣīṣī, [['Abd al-Hamīd ibn Turk]], [[Sind ibn Ali|Sind ibn 'Alī]], [[Sahl ibn Bishr|Sahl ibn Bišr]], and [[Sharaf al-Dīn al-Ṭūsī]].

[[Solomon Gandz]] has described Al-Khwarizmi as the father of Algebra:
{{blockquote|Al-Khwarizmi's algebra is regarded as the foundation and cornerstone of the sciences. In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers.<ref>[[Solomon Gandz|Gandz, Solomon]], The sources of al-Khwarizmi's algebra, Osiris, i (1936), 263–277</ref>}}

[[Victor J. Katz]] adds :
{{blockquote|The first true algebra text which is still extant is the work on al-jabr and al-muqabala by Mohammad ibn Musa al-Khwarizmi, written in Baghdad around 825.<ref>{{Cite journal|last=Katz|first=Victor J.|author-link= Victor J. Katz |title=Stages in the History of Algebra with Implications for Teaching|url=https://eclass.uoa.gr/modules/document/file.php/MATH104/20010-11/HistoryOfAlgebra.pdf|journal=VICTOR J.KATZ, University of the District of Columbia Washington DC, USA|pages=190|via=University of the District of Columbia Washington DC, USA|access-date=2017-10-07|archive-url=https://web.archive.org/web/20190327085930/https://eclass.uoa.gr/modules/document/file.php/MATH104/20010-11/HistoryOfAlgebra.pdf|archive-date=2019-03-27|url-status=dead}}</ref>}}

John J. O'Connor and [[Edmund F. Robertson]] wrote in the ''[[MacTutor History of Mathematics Archive]]'':
{{blockquote|Perhaps one of the most significant advances made by [[Mathematics in medieval Islam|Arabic mathematics]] began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed [[rational number]]s, [[irrational number]]s, geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before.<ref name=MacTutor>{{MacTutor|id=Al-Khwarizmi|title=Abu Ja'far Muhammad ibn Musa Al-Khwarizmi}}</ref>}}

[[Roshdi Rashed]] and Angela Armstrong write:
{{blockquote|Al-Khwarizmi's text can be seen to be distinct not only from the [[Babylonian mathematics|Babylonian tablets]], but also from [[Diophantus]]' ''[[Arithmetica]]''. It no longer concerns a series of [[problem solving|problems to be solved]], but an [[Rhetorical modes#Exposition|exposition]] which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study. On the other hand, the idea of an equation for its own sake appears from the beginning and, one could say, in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems.<ref>{{Cite book | last1=Rashed | first1=R. |author1-link=Roshdi Rashed | last2=Armstrong | first2=Angela | date=1994 | title=The Development of Arabic Mathematics | publisher=[[Springer Science+Business Media|Springer]] | isbn=978-0-7923-2565-9 | oclc=29181926 | pages=11–12 }}</ref>}}

According to Swiss-American historian of mathematics, [[Florian Cajori]], Al-Khwarizmi's algebra was different from the work of [[Indian mathematicians]], for Indians had no rules like the ''restoration'' and ''reduction''.<ref>{{Cite book|url=https://archive.org/details/ahistorymathema02cajogoog|title=A History of Mathematics|last=[[Florian Cajori]]|publisher=Macmillan|year=1919|page=[https://archive.org/details/ahistorymathema02cajogoog/page/n117 103]|quote=That it came from Indian source is impossible, for Hindus had no rules like "restoration" and "reduction". They were never in the habit of making all terms in an equation positive, as is done in the process of "restoration.}}</ref> Regarding the dissimilarity and significance of Al-Khwarizmi's algebraic work from that of Indian Mathematician [[Brahmagupta]], [[Carl Benjamin Boyer|Carl B. Boyer]] wrote: <blockquote>It is true that in two respects the work of al-Khowarizmi represented a retrogression from that of [[Diophantus]]. First, it is on a far more elementary level than that found in the Diophantine problems and, second, the algebra of al-Khowarizmi is thoroughly rhetorical, with none of the syncopation found in the Greek ''Arithmetica'' or in Brahmagupta's work. Even numbers were written out in words rather than symbols! It is quite unlikely that al-Khwarizmi knew of the work of Diophantus, but he must have been familiar with at least the astronomical and computational portions of Brahmagupta; yet neither al-Khwarizmi nor other Arabic scholars made use of syncopation or of negative numbers. Nevertheless, the ''Al-jabr'' comes closer to the elementary algebra of today than the works of either Diophantus or Brahmagupta, because the book is not concerned with difficult problems in indeterminant analysis but with a straight forward and elementary exposition of the solution of equations, especially that of second degree. The Arabs in general loved a good clear argument from premise to conclusion, as well as systematic organization – respects in which neither Diophantus nor the Hindus excelled.<ref>{{Cite book|url=https://archive.org/details/AHistoryOfMathematics|title=A History of Mathematics|last=Boyer |first=Carl Benjamin |author-link=Carl Benjamin Boyer |year=1968|page=[https://archive.org/details/AHistoryOfMathematics/page/n268 252]}}</ref></blockquote>

=== Arithmetic ===
[[File:Gregor Reisch, Margarita Philosophica, 1508 (1230x1615).png|thumb|upright=.8|Algorists vs. abacists, depicted in a sketch from 1508 CE]]
[[File:Dixit algorizmi.png|thumb|upright=.8|Page from a Latin translation, beginning with "Dixit algorizmi"]]
Al-Khwārizmī's second most influential work was on the subject of arithmetic, which survived in Latin translations but is lost in the original Arabic. His writings include the text ''kitāb al-ḥisāb al-hindī'' ('Book of Indian computation'{{refn|group=note|Some scholars translate the title ''al-ḥisāb al-hindī'' as "computation with Hindu numerals", but Arabic ''Hindī'' means 'Indian' rather than 'Hindu'. A. S. Saidan states that it should be understood as arithmetic done "in the Indian way", with Hindu-Arabic numerals, rather than as simply "Indian arithmetic". The Arab mathematicians incorporated their own innovations in their texts.<ref>{{citation |title=The Earliest Extant Arabic Arithmetic: Kitab al-Fusul fi al Hisab al-Hindi of Abu al-Hasan, Ahmad ibn Ibrahim al-Uqlidisi |first=A. S. |last=Saidan |journal=Isis |volume=57 |pages=475–490 |number=4 |date=Winter 1966 |publisher=The University of Chicago Press |jstor=228518|doi=10.1086/350163 |s2cid=143979243 }}</ref>}}), and perhaps a more elementary text, ''kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī'' ('Addition and subtraction in Indian arithmetic').{{sfn|Burnett|2017|p=39}}<ref>{{citation |last=Avari |first=Burjor |author-link=Burjor Avari |title=Islamic Civilization in South Asia: A history of Muslim power and presence in the Indian subcontinent |publisher=Routledge |year=2013 |isbn=978-0-415-58061-8 |url=https://books.google.com/books?id=hGHpVtQ8eKoC |pages=31–32 |access-date=29 September 2020 |archive-date=28 March 2023 |archive-url=https://web.archive.org/web/20230328222543/https://books.google.com/books?id=hGHpVtQ8eKoC |url-status=live }}</ref> These texts described algorithms on decimal numbers ([[Hindu–Arabic numeral system|Hindu–Arabic numerals]]) that could be carried out on a dust board. Called ''takht'' in Arabic (Latin: ''tabula''), a board covered with a thin layer of dust or sand was employed for calculations, on which figures could be written with a stylus and easily erased and replaced when necessary. Al-Khwarizmi's algorithms were used for almost three centuries, until replaced by [[Al-Uqlidisi]]'s algorithms that could be carried out with pen and paper.<ref>{{citation |first=Glen |last=Van Brummelen |author-link=Glen Van Brummelen |chapter=Arithmetic |editor=Thomas F. Glick |title=Routledge Revivals: Medieval Science, Technology and Medicine (2006): An Encyclopedia |chapter-url=https://books.google.com/books?id=zTQrDwAAQBAJ&pg=PA46 |year=2017 |publisher=Taylor & Francis |isbn=978-1-351-67617-5 |page=46 |access-date=5 May 2019 |archive-date=28 March 2023 |archive-url=https://web.archive.org/web/20230328222557/https://books.google.com/books?id=zTQrDwAAQBAJ&pg=PA46 |url-status=live }}</ref>

As part of 12th century wave of Arabic science flowing into Europe via translations, these texts proved to be revolutionary in Europe.<ref>{{citation |chapter=Al-Khwarizmi |editor=Thomas F. Glick |title=Routledge Revivals: Medieval Science, Technology and Medicine (2006): An Encyclopedia |chapter-url=https://books.google.com/books?id=zTQrDwAAQBAJ&pg=PA298 |year=2017 |publisher=Taylor & Francis |isbn=978-1-351-67617-5 |access-date=6 May 2019 |archive-date=28 March 2023 |archive-url=https://web.archive.org/web/20230328222552/https://books.google.com/books?id=zTQrDwAAQBAJ&pg=PA298 |url-status=live }}</ref>  Al-Khwarizmi's [[List of Latinised names|Latinized]] name, ''Algorismus'', turned into the [[algorism|name of method]] used for computations, and survives in the term "[[algorithm]]". It gradually replaced the previous abacus-based methods used in Europe.<ref>{{citation |first=Glen |last=Van Brummelen |author-link=Glen Van Brummelen |chapter=Arithmetic |editor=Thomas F. Glick |title=Routledge Revivals: Medieval Science, Technology and Medicine (2006): An Encyclopedia |chapter-url=https://books.google.com/books?id=zTQrDwAAQBAJ&pg=PA46 |year=2017 |publisher=Taylor & Francis |isbn=978-1-351-67617-5 |pages=46–47 |access-date=5 May 2019 |archive-date=28 March 2023 |archive-url=https://web.archive.org/web/20230328222557/https://books.google.com/books?id=zTQrDwAAQBAJ&pg=PA46 |url-status=live }}</ref>

Four Latin texts providing adaptions of Al-Khwarizmi's methods have survived, even though none of them is believed to be a literal translation:{{sfn|Burnett|2017|p=39}}
* ''Dixit Algorizmi'' (published in 1857 under the title ''Algoritmi de Numero Indorum''<ref name="Algorithmi">{{citation |chapter=Algoritmi de numero Indorum |title=Trattati D'Aritmetica |year=1857 |publisher=Tipografia delle Scienze Fisiche e Matematiche |location=Rome |pages=1– |chapter-url=https://books.google.com/books?id=1J9GAAAAcAAJ&pg=PA1 |access-date=6 May 2019 |archive-date=28 March 2023 |archive-url=https://web.archive.org/web/20230328222535/https://books.google.com/books?id=1J9GAAAAcAAJ&pg=PA1 |url-status=live }}</ref>)<ref name="Thus spake" />
* ''Liber Alchoarismi de Practica Arismetice''
* ''Liber Ysagogarum Alchorismi''
* ''Liber Pulveris''
''Dixit Algorizmi'' ('Thus spake Al-Khwarizmi') is the starting phrase of a manuscript in the University of Cambridge library, which is generally referred to by its 1857 title ''Algoritmi de Numero Indorum''. It is attributed to the [[Adelard of Bath]], who had translated the astronomical tables in 1126. It is perhaps the closest to Al-Khwarizmi's own writings.<ref name="Thus spake">{{citation |first1=John N. |last1=Crossley |first2=Alan S. |last2=Henry |journal=Historia Mathematica |volume=17 |issue=2 |pages=103–131 |year=1990 |title=Thus Spake al-Khwārizmī: A Translation of the Text of Cambridge University Library Ms. Ii.vi.5 |doi=10.1016/0315-0860(90)90048-I|doi-access=free }}</ref>

Al-Khwarizmi's work on arithmetic was responsible for introducing the [[Arabic numerals]], based on the [[Hindu–Arabic numeral system]] developed in [[Indian mathematics]], to the Western world. The term "algorithm" is derived from the [[algorism]], the technique of performing arithmetic with Hindu-Arabic numerals developed by al-Khwārizmī. Both "algorithm" and "algorism" are derived from the [[List of Latinised names|Latinized forms]] of al-Khwārizmī's name, ''Algoritmi'' and ''Algorismi'', respectively.<ref>{{Cite web|url=https://earthobservatory.nasa.gov/images/91544/how-algorithm-got-its-name|title=How Algorithm Got Its Name|date=8 January 2018|website=earthobservatory.nasa.gov}}</ref>

=== Astronomy ===
{{Further|Astronomy in the medieval Islamic world}}
[[File:Corpus Christ College MS 283 (1).png|thumb|upright=.7|Page from ''Corpus Christi College MS 283'', a Latin translation of al-Khwārizmī's ''Zīj'']]

Al-Khwārizmī's [[Zij as-Sindhind|{{transliteration|ar|Zīj as-Sindhind}}]]<ref name="toomer" /> ({{langx|ar|زيج السند هند}}, "[[zij|astronomical tables]] of ''[[Siddhanta#Astronomy|Siddhanta]]''"<ref name="Thurston1996">{{citation|last=Thurston|first=Hugh|title=Early Astronomy|url=https://books.google.com/books?id=rNpHjqxQQ9oC&pg=PP204|year=1996|publisher=Springer Science & Business Media|isbn=978-0-387-94822-5|pages=204–}}</ref>) is a work consisting of approximately 37 chapters on calendrical and astronomical calculations and 116 tables with calendrical, astronomical and astrological data, as well as a table of sine values.  This is the first of many Arabic ''[[Zij]]es'' based on the [[Indian astronomy|Indian astronomical]] methods known as the ''sindhind''.<ref name=Kennedy-1956>{{harvnb|Kennedy|1956|pp= 26–29}}</ref>  The word Sindhind is a corruption of the [[Sanskrit]] ''Siddhānta'', which is the usual designation of an astronomical textbook. In fact, the mean motions in the tables of al-Khwarizmi are derived from those in the "corrected Brahmasiddhanta" ([[Brāhmasphuṭasiddhānta|Brahmasphutasiddhanta]]) of [[Brahmagupta]].<ref>{{Cite book|last=van der Waerden|first=Bartel Leendert |author-link=Bartel Leendert van der Waerden |url=https://www.springer.com/gp/book/9783642516016|title=A History of Algebra: From al-Khwārizmī to Emmy Noether|date=1985|publisher=Springer-Verlag|isbn=978-3-642-51601-6|location=Berlin Heidelberg|pages=10|language=en|access-date=22 June 2021|archive-date=24 June 2021|archive-url=https://web.archive.org/web/20210624203930/https://www.springer.com/gp/book/9783642516016|url-status=live}}</ref>

The work contains tables for the movements of the [[sun]], the [[moon]] and the five [[planet]]s known at the time. This work marked the turning point in [[Islamic astronomy]]. Hitherto, Muslim astronomers had adopted a primarily research approach to the field, translating works of others and learning already discovered knowledge.<!-- Al-Khwarizmi's work marked the beginning of non-traditional methods of study and calculations.<ref>{{Harv|Dallal|1999|p=163}}</ref> ?? -->

The original Arabic version (written {{Circa|820}}) is lost, but a version by the Spanish astronomer [[Maslama al-Majriti]] ({{Circa|1000}}) has survived in a Latin translation, presumably by [[Adelard of Bath]] (26 January 1126).<ref>{{harvnb|Kennedy|1956|p=128}}</ref> The four surviving manuscripts of the Latin translation are kept at the Bibliothèque publique (Chartres), the Bibliothèque Mazarine (Paris), the Biblioteca Nacional (Madrid) and the Bodleian Library (Oxford).
<!-- FAILED VERIFICATION!
Al-Khwarizmi made important improvements to the theory and construction of [[sundial]]s, which he inherited from his [[Indian astronomy|Indian]] and [[Hellenistic civilization|Hellenistic]] predecessors. He made tables for these instruments which considerably shortened the time needed to make specific calculations. His sundial was universal and could be observed from anywhere on the Earth. From then on, sundials were frequently placed on mosques to determine the [[Salah|time of prayer]].<ref>{{Harv|King|1999a|pp=168–169}}</ref> The shadow square, an instrument used to determine the linear height of an object, in conjunction with the [[alidade]] for angular observations, was invented by al-Khwārizmī in ninth-century [[Baghdad]].<ref>{{cite journal | last1 = King | first1 = David A. |author-link= David A. King (historian) | year = 2002 | title = A Vetustissimus Arabic Text on the Quadrans Vetus | journal = Journal for the History of Astronomy | volume = 33 | pages = 237–255 [238–239] |bibcode = 2002JHA....33..237K | doi=10.1177/002182860203300302}}</ref>{{Failed verification|date=April 2010}}

The first [[Quadrant (instrument)|quadrants]] and [[mural instrument]]s were invented by al-Khwārizmī in ninth century Baghdad.<ref name=King>[[David A. King (historian)|King, David A.]], "Islamic Astronomy", in Christopher Walker (1999), ed., ''Astronomy before the telescope'', pp.&nbsp;167–168. [[British Museum]] Press. {{isbn|0-7141-2733-7}}.</ref>{{Failed verification|date=April 2010}} The sine quadrant, invented by al-Khwārizmī, was used for astronomical calculations.<ref name=King-2002 />{{Failed verification|date=April 2010}} The first horary [[Quadrant (instrument)|quadrant]] for specific [[latitude]]s, was invented by al-Khwārizmī in Baghdad, then center of the development of quadrants.<ref name=King-2002 />{{Failed verification|date=April 2010}} It was used to determine time (especially the times of prayer) by observations of the Sun or stars.<ref>{{Harv|King|1999a|pp=167–168}}</ref> The ''Quadrans Vetus'' was a universal horary quadrant, an ingenious mathematical device invented by al-Khwārizmī in the ninth century and later known as the ''Quadrans Vetus'' (''Old Quadrant'') in medieval Europe from the 13th century. It could be used for any [[latitude]] on Earth and at any time of the year to determine the time in hours from the altitude of the Sun. This was the second most widely used astronomical instrument during the [[Middle Ages]] after the [[astrolabe]]. One of its main purposes in the Islamic world was to determine the times of [[Salah]].<ref name=King-2002>{{Harv|King|2002|pp=237–238}}</ref>{{Failed verification|date=April 2010}}-->

=== Trigonometry ===
Al-Khwārizmī's ''Zīj as-Sindhind'' contained tables for the [[trigonometric functions]] of sines and cosine.<ref name=Kennedy-1956 /> A related treatise on [[spherical trigonometry]] is attributed to him.<ref name=MacTutor />

Al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents.<ref name="Sesiano">Jacques Sesiano, "Islamic mathematics", p. 157, in {{Cite book |title=Mathematics Across Cultures: The History of Non-western Mathematics |editor1-first=Helaine |editor1-last=Selin |editor1-link=Helaine Selin |editor2-first=Ubiratan |editor2-last=D'Ambrosio |editor2-link=Ubiratan D'Ambrosio |year=2000 |publisher=[[Springer Science+Business Media]] |isbn=978-1-4020-0260-1}}</ref><ref name="Britannica">{{cite encyclopedia |title=trigonometry |url=https://www.britannica.com/EBchecked/topic/605281/trigonometry |encyclopedia=[[Encyclopædia Britannica]] |access-date=21 July 2008 |archive-date=6 July 2008 |archive-url=https://web.archive.org/web/20080706200811/http://www.britannica.com/EBchecked/topic/605281/trigonometry |url-status=live }}</ref>

=== Geography ===
[[File:World map by Al-Khwarizmi.svg|thumb|upright=1.3|Gianluca Gorni's reconstruction of the section of al-Khwārizmī's world map concerning the Indian Ocean. The majority of the placenames used by al-Khwārizmī match those of Ptolemy, [[Henricus Martellus Germanus|Martellus]] and [[Martin Behaim|Behaim]]. The general shape of the coastline is the same between [[Taprobane]] and [[Cattigara]]. The [[Dragon's Tail (peninsula)|Dragon's Tail]], or the eastern opening of the Indian Ocean, which does not exist in Ptolemy's description, is traced in very little detail on al-Khwārizmī's map, although is clear and precise on the Martellus map and on the later Behaim version.]]
[[File:PtolemyWorldMap.jpg|thumb|A [[Ptolemy's world map|15th-century version]] of [[Claudius Ptolemy|Ptolemy]]'s [[Ptolemy's Geography|''Geography'']] for comparison]]
[[File:Earliest extant map of the Nile, in al-Khwārazmī’s Kitāb ṣūrat al- arḍ.jpg|thumb|Earliest extant map of the Nile, in Al-Khwārazmī’s Kitāb ṣūrat al- arḍ.]]
Al-Khwārizmī's third major work is his {{transliteration|ar|Kitāb Ṣūrat al-Arḍ}} ({{langx|ar|كتاب صورة الأرض}}, "Book of the Description of the Earth"),{{refn|The full title is "The Book of the Description of the Earth, with its Cities, Mountains, Seas, All the Islands and the Rivers, written by Abu Ja'far Muhammad ibn Musa al-Khwārizmī, according to the Geographical Treatise written by Ptolemy the Claudian",<!--sic--> although due to ambiguity in the word ''surah'' it could also be understood as meaning "The Book of the Image of the Earth" or even "The Book of the Map of the World".}} also known as his ''Geography'', which was finished in 833. It is a major reworking of [[Ptolemy]]'s second-century ''[[Geography (Ptolemy)|Geography]]'', consisting of a list of 2402 coordinates of cities and other geographical features following a general introduction.<ref>{{cite web|access-date=30 May 2008|url=http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Cartography.html|title=The history of cartography|publisher=[[GAP computer algebra system]]|url-status=dead|archive-url=https://web.archive.org/web/20080524092016/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Cartography.html|archive-date=24 May 2008}}</ref>

There is one surviving copy of {{transliteration|ar|Kitāb Ṣūrat al-Arḍ}}, which is kept at the [[Strasbourg University Library]].<ref>{{Cite web |title=Consultation |url=https://archivesetmanuscrits.bnf.fr/ark:/12148/cc96697g/ca19904036 |access-date=2024-08-27 |website=archivesetmanuscrits.bnf.fr}}</ref><ref>{{Cite book |last=al-Ḫwarizmī |first=Muḥammad Ibn Mūsā |url=https://books.google.com/books?id=uL8BQwAACAAJ |title=Das Kitāb ṣūrat al-arḍ des Abū Ǧaʻfar Muḥammad Ibn Mūsā al-Ḫuwārizmī |date=1926 |language=ar}}</ref> A Latin translation is at the [[Biblioteca Nacional de España]] in Madrid.<ref>{{cite book
 | title = The Man of Numbers: Fibonacci's Arithmetic Revolution | author = Keith J. Devlin
 | year = 2012
 | publisher = Bloomsbury
 | format = Paperback | isbn = 9781408822487 | page = 55|url=https://books.google.com/books?id=RFMmPGa0cisC&q=Kit%C4%81b+%E1%B9%A2%C5%ABrat+al-Ar%E1%B8%8D+Biblioteca+Nacional+de+Espa%C3%B1a+-wikipedia}}</ref> The book opens with the list of [[latitudes]] and [[longitudes]], in order of "weather zones", that is to say in blocks of latitudes and, in each weather zone, by order of longitude. As [[Paul Gallez]] notes, this system allows the deduction of many latitudes and longitudes where the only extant document is in such a bad condition, as to make it practically illegible. Neither the Arabic copy nor the Latin translation include the map of the world; however, Hubert Daunicht was able to reconstruct the missing map from the list of coordinates. Daunicht read the latitudes and longitudes of the coastal points in the manuscript, or deduced them from the context where they were not legible. He transferred the points onto [[graph paper]] and connected them with straight lines, obtaining an approximation of the coastline as it was on the original map. He did the same for the rivers and towns.<ref>Daunicht</ref>

Al-Khwārizmī corrected Ptolemy's gross overestimate for the length of the [[Mediterranean Sea]]<ref name=Kennedy-188>Edward S. Kennedy, ''Mathematical Geography'', p.&nbsp;188, in {{Harv|Rashed|Morelon|1996|pp=185–201}}</ref> from the [[Canary Islands]] to the eastern shores of the Mediterranean; Ptolemy overestimated it at 63 degrees of [[longitude]], while al-Khwārizmī almost correctly estimated it at nearly 50 degrees of longitude. He "depicted the [[Atlantic Ocean|Atlantic]] and Indian Oceans as [[Ocean|open bodies of water]], not land-locked seas as Ptolemy had done."<ref name=Covington>{{Cite journal|first=Richard|last=Covington|journal=Saudi Aramco World, May–June 2007|date=2007|pages=17–21|url=http://www.saudiaramcoworld.com/issue/200703/the.third.dimension.htm|title=The Third Dimension|access-date=6 July 2008|archive-url=https://web.archive.org/web/20080512022044/http://www.saudiaramcoworld.com/issue/200703/the.third.dimension.htm|archive-date=12 May 2008|url-status=dead}}</ref> Al-Khwārizmī's [[Prime Meridian]] at the [[Fortunate Isles]] was thus around 10° east of the line used by Marinus and Ptolemy. Most medieval Muslim gazetteers continued to use al-Khwārizmī's prime meridian.<ref name=Kennedy-188 />

=== Jewish calendar ===
Al-Khwārizmī wrote several other works including a treatise on the [[Hebrew calendar]], titled {{transliteration|ar|Risāla fi istikhrāj ta'rīkh al-yahūd}} ({{langx|ar|رسالة في إستخراج تأريخ اليهود}}, "Extraction of the Jewish Era"). It describes the [[Metonic cycle]], a 19-year intercalation cycle; the rules for determining on what day of the week the first day of the month [[Tishrei]] shall fall; calculates the interval between the [[Anno Mundi]] or Jewish year and the [[Seleucid era]]; and gives rules for determining the mean longitude of the sun and the moon using the [[Hebrew calendar]]. Similar material is found in the works of [[Al-Bīrūnī]] and [[Maimonides]].<ref name="toomer" /> <!-- Folkerts / More in Kenedy / Only Sezgin mentions "risala fi" -->

=== Other works ===
[[Ibn al-Nadim]]'s {{transliteration|ar|Al-Fihrist}}, an index of Arabic books, mentions al-Khwārizmī's {{transliteration|ar|Kitāb al-Taʾrīkh}} ({{langx|ar|كتاب التأريخ}}), a book of annals. No direct manuscript survives; however, a copy had reached [[Nusaybin]] by the 11th century, where its [[metropolitan bishop]], Mar [[Elias bar Shinaya]], found it. Elias's chronicle quotes it from "the death of the Prophet" through to 169 AH, at which point Elias's text itself hits a lacuna.<ref>{{cite book |author= LJ Delaporte |title=Chronographie de Mar Elie bar Sinaya |date=1910 |page=xiii}}</ref>

Several Arabic manuscripts in Berlin, Istanbul, Tashkent, Cairo and Paris contain further material that surely or with some probability comes from al-Khwārizmī. The Istanbul manuscript contains a paper on sundials; the ''Fihrist'' credits al-Khwārizmī with {{transliteration|ar|Kitāb ar-Rukhāma(t)}} ({{langx|ar|كتاب الرخامة}}).<!-- This is likely an error in the Fihirst (Dunlop) --> Other papers, such as one on the determination of the direction of [[Mecca]], are on the [[spherical astronomy]].

Two texts deserve special interest on the [[morning width]] ({{transliteration|ar|Ma'rifat sa'at al-mashriq fī kull balad}}) and the determination of the [[azimuth]] from a height <!-- Bestimmung des Azimuts aus der Höhe --> ({{transliteration|ar|Ma'rifat al-samt min qibal al-irtifā'}}). <!-- see Rosenfeld 1993 --> He wrote two books on using and constructing [[astrolabe]]s.

== Honours ==
[[Image:1983 CPA 5426 (1).png|A Soviet [[commemorative stamp|postage stamp]] issued 6 September 1983, commemorating al-Khwārizmī's (approximate) 1200th birthday|thumb|upright=.8]]
* [[Al-Khwarizmi (crater)]]&nbsp;— A crater on the far side of the Moon. <ref>{{cite journal |last1=El-Baz |first1=Farouk |title=Al-Khwarizmi: A New-Found Basin on the Lunar Far Side |journal=Science |date=1973 |volume=180 |issue=4091 |pages=1173–1176 |url=https://www.jstor.org/stable/1736378|doi=10.1126/science.180.4091.1173|jstor=1736378|pmid=17743602 |bibcode=1973Sci...180.1173E |s2cid=10623582 }} NASA Portal: [https://history.nasa.gov/afj/ap11fj/photos/43-t.html Apollo 11, Photography Index].
</ref>
*[[13498 Al Chwarizmi]] — Main-belt Asteroid, Discovered 1986 Aug 6 by E. W. Elst and V. G. Ivanova at Smolyan.<ref>{{Cite web|url=https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/?sstr=13498|title=Small-Body Database Lookup|website=ssd.jpl.nasa.gov}}</ref>
*[[11156 Al-Khwarismi]] — Main-belt Asteroid, Discovered 1997 Dec 31 by P. G. Comba at Prescott.<ref>{{Cite web|url=https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/?sstr=11156|title=Small-Body Database Lookup|website=ssd.jpl.nasa.gov}}</ref>

== Notes ==
{{reflist|group=note}}

== References ==
{{Reflist}}

===Sources===
{{Refbegin|30em}}
* {{Cite journal|last=Arndt|first=A. B.|title=Al-Khwarizmi |journal=The Mathematics Teacher |date=December 1983 |pages=668–670|volume=76|issue=9 |doi=10.5951/MT.76.9.0668|jstor=27963784}}
* {{cite book|first=Carl B.|last=Boyer|author-link=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second|publisher=John Wiley & Sons, Inc.|date=1991|chapter=The Arabic Hegemony|isbn=978-0-471-54397-8|url=https://archive.org/details/historyofmathema00boye}}
* {{citation |first=Charles |last=Burnett |chapter=Arabic Numerals |editor=Thomas F. Glick |title=Routledge Revivals: Medieval Science, Technology and Medicine (2006): An Encyclopedia |chapter-url=https://books.google.com/books?id=zTQrDwAAQBAJ&pg=PA39 |year=2017 |publisher=Taylor & Francis |isbn=978-1-351-67617-5 |access-date=5 May 2019 |archive-date=28 March 2023 |archive-url=https://web.archive.org/web/20230328222536/https://books.google.com/books?id=zTQrDwAAQBAJ&pg=PA39 |url-status=live }}
* {{Cite book|last=Daffa|first=Ali Abdullah al-|author-link=Ali Abdullah Al-Daffa|title=The Muslim contribution to mathematics|date=1977|publisher=[[Croom Helm]]|location=London|isbn= 978-0-85664-464-1}}
* {{cite journal |last1=Dunlop |first1=Douglas Morton |title=''Muḥammad b. Mūsā al-Khwārizmī'' |journal=The Journal of the Royal Asiatic Society of Great Britain and Ireland |date=1943 |volume=2 |issue=3–4 |pages=248–250 |doi=10.1017/S0035869X00098464 |jstor=25221920 |s2cid=161841351 |url=https://www.jstor.org/stable/25221920 |access-date=24 June 2021 |archive-date=25 June 2021 |archive-url=https://web.archive.org/web/20210625061227/https://www.jstor.org/stable/25221920 |url-status=live }}
* {{cite journal |last1=Kennedy |first1=E. S. |title=A Survey of Islamic Astronomical Tables |journal=Transactions of the American Philosophical Society |year=1956 |volume=46 |issue=2 |pages=123–177 |doi=10.2307/1005726 |jstor=1005726 |hdl=2027/mdp.39076006359272 |url=https://www.jstor.org/stable/1005726 |hdl-access=free |access-date=24 June 2021 |archive-date=4 June 2021 |archive-url=https://web.archive.org/web/20210604153154/https://www.jstor.org/stable/1005726 |url-status=live }}
*{{Citation
 |last2=Morelon
 |first2=Régis
 |last1=Rashed
 |first1=Roshdi
 |author1-link=Roshdi Rashed
 |year=1996
 |title=Encyclopedia of the History of Arabic Science
 |url=https://books.google.com/books?id=dIWtmfwvItkC
 |volume=1
 |publisher=[[Routledge]]
 |isbn=0-415-12410-7
 }}
* {{Cite book|last=Struik|first=Dirk Jan|author-link=Dirk Jan Struik|title=A Concise History of Mathematics|date=1987|isbn=978-0-486-60255-4|edition=4th|publisher=[[Dover Publications]]|url-access=registration|url=https://archive.org/details/concisehistoryof0000stru_m6j1}}
* {{cite encyclopedia
  | last = Toomer
  | first = Gerald
  | author-link = Gerald Toomer
  | title = Al-Khwārizmī, Abu Ja'far Muḥammad ibn Mūsā
  | encyclopedia = [[Dictionary of Scientific Biography]]
  | volume = 7
  | editor = Gillispie, Charles Coulston
  | publisher = Charles Scribner's Sons
  | location = New York
  | date = 1990
  | isbn = 978-0-684-16962-0
  | url = http://www.encyclopedia.com/doc/1G2-2830902300.html
  | access-date = 31 December 2010
  | archive-date = 2 July 2016
  | archive-url = https://web.archive.org/web/20160702040538/http://www.encyclopedia.com/doc/1G2-2830902300.html
  | url-status = live
  }}
{{Refend}}

== Further reading ==
{{Refbegin|30em}}

=== Biographical ===
* [[Sonja Brentjes|Brentjes, Sonja]] (2007). "[http://islamsci.mcgill.ca/RASI/BEA/Khwarizmi_BEA.htm Khwārizmī: Muḥammad ibn Mūsā al-Khwārizmī] {{Webarchive|url=https://web.archive.org/web/20110706185327/http://islamsci.mcgill.ca/RASI/BEA/Khwarizmi_BEA.htm |date=6 July 2011 }}" in Thomas Hockey et al.(eds.). ''[[The Biographical Encyclopedia of Astronomers]]'', Springer Reference. New York: Springer, 2007, pp.&nbsp;631–633. ([http://islamsci.mcgill.ca/RASI/BEA/Khwarizmi_BEA.pdf PDF version] {{Webarchive|url=https://web.archive.org/web/20120114103045/http://islamsci.mcgill.ca/RASI/BEA/Khwarizmi_BEA.pdf |date=14 January 2012 }})
* [[Jan Hogendijk|Hogendijk, Jan P.]], [http://www.alkhwarizmi.nl/ Muhammad ibn Musa (Al-)Khwarizmi (c. 780–850 CE)] {{Webarchive|url=https://web.archive.org/web/20180203102909/http://alkhwarizmi.nl/ |date=3 February 2018 }} – bibliography of his works, manuscripts, editions and translations.
* {{MacTutor Biography|id=Al-Khwarizmi|title=Abu Ja'far Muhammad ibn Musa Al-Khwarizmi}}
* Sezgin, F., ed.,  ''Islamic Mathematics and Astronomy'',  Frankfurt: Institut für Geschichte der arabisch-islamischen Wissenschaften, 1997–99. <!-- This is a collection of (mostly) reprints, consisting of 112 volumes to date. Practically all the literature on Islamic mathematics published before 1960 will be reprinted in these volumes. The volumes are compiled thematically, for example vols. 1–4 are about Al-Khwarizmi, vols. 14–20 on Euclid in the Arabic tradition, vols. 21–22 on Tabit ibn Qurra, vol. 23 on Abu Kamil, vols. 24–25 on Ibn Yunis, vols. 32–36 on al-Biruni, etc. -->

=== Algebra ===
* {{cite journal |last1=Gandz |first1=Solomon |author1-link=Solomon Gandz |title=''The Origin of the Term "Algebra'' |journal=The American Mathematical Monthly |date=November 1926 |volume=33 |issue=9 |pages=437–440 |doi=10.2307/2299605 |jstor=2299605 |url=https://www.jstor.org/stable/2299605 |access-date=24 June 2021 |archive-date=25 June 2021 |archive-url=https://web.archive.org/web/20210625055507/https://www.jstor.org/stable/2299605 |url-status=live }}
* {{cite journal |last1=Gandz |first1=Solomon |author1-link=Solomon Gandz |title=''The Sources of al-Khowārizmī's Algebra'' |journal=Osiris |date=1936 |volume=1 |issue=1 |pages=263–277 |doi=10.1086/368426 |jstor=301610 |s2cid=60770737 |url=https://www.jstor.org/stable/301610 |access-date=24 June 2021 |archive-date=25 June 2021 |archive-url=https://web.archive.org/web/20210625080209/https://www.jstor.org/stable/301610 |url-status=live }}
* {{cite journal |last1=Gandz |first1=Solomon |author1-link=Solomon Gandz |title=''The Algebra of Inheritance: A Rehabilitation of Al-Khuwārizmī'' |journal=Osiris |date=1938 |volume=5 |issue=5 |pages=319–391 |doi=10.1086/368492 |jstor=301569 |s2cid=143683763 |url=https://www.jstor.org/stable/301569 |access-date=24 June 2021 |archive-date=25 June 2021 |archive-url=https://web.archive.org/web/20210625054923/https://www.jstor.org/stable/301569 |url-status=live }}
* {{cite journal |last1=Hughes |first1=Barnabas |title=Gerard of Cremona's Translation of al-Khwārizmī's al-Jabr, A Critical Edition |journal=Mediaeval Studies |date=1986 |volume=48 |pages=211–263 |doi=10.1484/J.MS.2.306339 |url=https://doi.org/10.1484/J.MS.2.306339 }}
* Hughes, Barnabas. ''Robert of Chester's Latin translation of al-Khwarizmi's al-Jabr: A new critical edition''. In Latin. F. Steiner Verlag Wiesbaden (1989). {{isbn|3-515-04589-9}}.
* {{cite book|first=L.C.|last=Karpinski|author-link=L. C. Karpinski|title=Robert of Chester's Latin Translation of the Algebra of Al-Khowarizmi: With an Introduction, Critical Notes and an English Version|date=1915|publisher=The Macmillan Company|url=https://library.albany.edu/preservation/brittlebooks|access-date=21 May 2020|archive-date=24 September 2020|archive-url=https://web.archive.org/web/20200924032358/https://library.albany.edu/preservation/brittlebooks|url-status=live}}
* {{cite book |last1=Rosen |first1=Fredrick |title=The Algebra of Mohammed Ben Musa |date=1831 |location=London |url=https://archive.org/details/algebraofmohamme00khuwuoft }}

=== Astronomy ===
* {{cite book|title=Commentary on the Astronomical Tables of Al-Khwarizmi: By Ibn Al-Muthanna|first=B.R.|last=Goldstein|author-link=B. R. Goldstein|publisher=Yale University Press|date=1968|isbn=978-0-300-00498-4}}
* {{cite journal |last1=Hogendijk |first1=Jan P. |author1-link= Jan Hogendijk |title=Al-Khwārizmī's Table of the "Sine of the Hours" and the Underlying Sine Table |journal=Historia Scientiarum |date=1991 |volume=42 |pages=1–12 |url=http://www.jphogendijk.nl/publ.html#English |access-date=24 June 2021 |archive-date=7 May 2021 |archive-url=https://web.archive.org/web/20210507023229/http://www.jphogendijk.nl/publ.html#English |url-status=live }} (Hogendijk's homepage. Publication in English, no. 25).
* {{cite book |last1=King |first1=David A. |author-link=David A. King (historian) |title=Al-Khwārizmī and New Trends in Mathematical Astronomy in the Ninth Century |date=1983 |publisher=Hagop Kevorkian Center for Near Eastern Studies: Occasional Papers on the Near East 2 |location=New York University |url=https://davidaking.academia.edu/research#muslimastronomers |access-date=24 June 2021 |archive-date=25 June 2021 |archive-url=https://web.archive.org/web/20210625030900/https://davidaking.academia.edu/research#muslimastronomers |url-status=live }} (Description and analysis of seven recently discovered minor works related to al-Khwarizmi).
* {{cite book|last=Neugebauer|first=Otto|author-link=Otto Neugebauer|title=The Astronomical Tables of al-Khwarizmi|date=1962}}
* {{cite book |last1=Rosenfeld |first1=Boris A. |editor1-last=Folkerts |editor1-first=Menso |editor2-last=Hogendijk |editor2-first=Jan P. |editor2-link=Jan Hogendijk |title=Vestigia Mathematica: Studies in Medieval and Early Modern Mathematics in Honour of H.L.L. Busard |date=1993 |publisher=Brill |location=Leiden |isbn=978-90-5183-536-6 |pages=305–308 |chapter='Geometric trigonometry' in treatises of al-Khwārizmī, al-Māhānī and Ibn al-Haytham}}
* {{cite book |last1=Van Dalen |first1=Benno |editor1-last=Casulleras |editor1-first=Josep |editor2-last=Samsó |editor2-first=Julio |title=From Baghdad to Barcelona, Studies on the Islamic Exact Sciences in Honour of Prof. Juan Vernet |date=1996 |publisher=Instituto Millás Vallicrosa de Historia de la Ciencia Arabe |location=Barcelona |pages=195–252 |url=http://www.bennovandalen.de/Publications/publications.html |chapter=al-Khwârizmî's Astronomical Tables Revisited: Analysis of the Equation of Time |chapter-url=http://www.bennovandalen.de/Publications/publications.html |access-date=24 June 2021 |archive-date=24 June 2021 |archive-url=https://web.archive.org/web/20210624203439/http://www.bennovandalen.de/Publications/publications.html |url-status=live }} (Van Dalen's homepage. List of Publications, Articles – no. 5).

=== Jewish calendar ===
* {{cite journal|last=Kennedy|first=E. S.|author-link=Edward Stewart Kennedy|title=Al-Khwārizmī on the Jewish Calendar|date=1964|journal=[[Scripta Mathematica]]|volume=27|pages=55–59}} <!-- reprinted in Studies in the Islamic Exact Sciences. Beirut 1983, 661–665 -->
{{Refend}}

==External links==
{{Wikiquote|al-Khwārizmī}}
*{{Commons category-inline|Muhammad ibn Musa al-Khwarizmi}}
*[https://archivesetmanuscrits.bnf.fr/ark:/12148/cc96697g/ca19904036 Earliest Manuscript of Kitab Surat al-Ard] in the [[National Academic Library (Strasbourg)|Strasbourg National Library]]

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{{People of Khorasan}}
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{{DEFAULTSORT:Khwarizmi, Muhammad ibn Musa}}
[[Category:780s births]]
[[Category:850 deaths]]
[[Category:8th-century Arabic-language writers]]
[[Category:8th-century astrologers]]
[[Category:8th-century Iranian astronomers]]
[[Category:8th-century people from the Abbasid Caliphate]]
[[Category:9th-century Arabic-language writers]]
[[Category:9th-century astrologers]]
[[Category:9th-century cartographers]]
[[Category:9th-century geographers]]
[[Category:9th-century inventors]]
[[Category:9th-century Iranian astronomers]]
[[Category:9th-century people from the Abbasid Caliphate]]
[[Category:9th-century Iranian mathematicians]]
[[Category:Astronomers from the Abbasid Caliphate]]
[[Category:Geographers from the Abbasid Caliphate]]
[[Category:Inventors of the medieval Islamic world]]
[[Category:Mathematicians from the Abbasid Caliphate]]
[[Category:Mathematicians who worked on Islamic inheritance]]
[[Category:Medieval Iranian astrologers]]
[[Category:Medieval Iranian geographers]]
[[Category:People from Khwarazm]]
[[Category:People from Xorazm Region]]
[[Category:Transoxanian Islamic scholars]]
[[Category:Medieval Iranian physicists]]
[[Category:Scientists who worked on qibla determination]]
[[Category:Writers about religion and science]]