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Al-Khwarizmi

October 16, 2023

For other uses, see Al-Khwarizmi (disambiguation).

Muḥammad ibn Mūsā al-Khwārizmī[note 1] (Arabic: محمد بن موسى الخوارزمي; c. 780 – c. 850), or al-Khwarizmi, was an Abbasid polymath from Khwarazm, who produced vastly influential works in mathematics, astronomy, and geography. Around 820 CE, he was appointed as the astronomer and head of the library of the House of Wisdom in Baghdad.[6]: 14 

Muḥammad ibn Mūsā al-Khwārizmī
محمد بن موسى الخوارزمي
Woodcut panel depicting al-Khwarizmi
Bornc. 780
DiedAfter 847[2][3] (aged c. 70)
Academic work
EraIslamic Golden Age
(Abbasid era)
Main interestsMathematics, astronomy, geography
Notable works
Notable ideasTreatises on algebra and Hindu–Arabic numeral system
InfluencedAbu Kamil[1]

Al-Khwarizmi's popularizing treatise on algebra (The Compendious Book on Calculation by Completion and Balancing, c. 813–833 CE[7]: 171 ) presented the first systematic solution of linear and quadratic equations. One of his principal achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications.[6]: 14  Because he was the first 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),[8] he has been described as the father[9][10][11] or founder[12][13] of algebra. The term algebra itself comes from the title of his book (the word al-jabr meaning "completion" or "rejoining").[14] His name gave rise to the terms algorism and algorithm,[15][16] the Spanish, Italian, and Portuguese terms algoritmo, and the Spanish guarismo[17] and Portuguese algarismo, both meaning "digit".[18]

In the 12th century, Latin translations of his textbook on arithmetic (Algorithmo de Numero Indorum) which codified the various Indian numerals, introduced the decimal positional number system to the Western world.[19] The Compendious Book on Calculation by Completion and Balancing, translated into Latin by Robert of Chester in 1145, was used until the sixteenth century as the principal mathematical text-book of European universities.[20][21][22][23]

In addition to his best-known works, he revised Ptolemy's Geography, listing the longitudes and latitudes of various cities and localities.[24]: 9  He further produced a set of astronomical tables and wrote about calendaric works, as well as the astrolabe and the sundial.[25] He also made important contributions to trigonometry, producing accurate sine and cosine tables, and the first table of tangents.[citation needed]

Life

 
Monument to Muhammad ibn Musa al-Khwarizmi at Ciudad Universitaria of Madrid

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.[26][27][28] Of Persian stock,[29][26][30][31][32] his name means 'the native of Khwarazm', a region that was part of Greater Iran,[33] and is now part of Turkmenistan, and Uzbekistan.[34]

Muhammad ibn Jarir al-Tabari gives his name as Muḥammad ibn Musá al-Khwārizmī al-Majūsī al-Quṭrubbullī (محمد بن موسى الخوارزميّ المجوسـيّ القطربّـليّ). The epithet al-Qutrubbulli could indicate he might instead have come from Qutrubbul (Qatrabbul),[35] near Baghdad. However, Rashed denies this:[36]

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 'و' for the conjunction '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, 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 affirms his nisba to Qutrubul, noting that he was called al-Khwārizmī al-Qutrubbulli because he was born just outside of Baghdad.[37]

Regarding al-Khwārizmī's religion, Toomer writes:[38]

Another epithet given to him by al-Ṭabarī, "al-Majūsī," would seem to indicate that he was an adherent of the old 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.

Ibn al-Nadīm's Kitāb 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. He worked in the House of Wisdom established by the Abbasid Caliph al-Ma'mūn, where he studied the sciences and mathematics, including the translation of Greek and Sanskrit scientific manuscripts. He was also a historian who is frequently cited by the likes of al-Tabari and Ibn Abi Tahir.[39]

During the reign of al-Wathiq, he is said to have been involved in the first of two embassies to the Khazars.[40]

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.[41]

Contributions

 
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, "The Compendious Book on Calculation by Completion and Balancing".[42]

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. It was translated into Latin as Algoritmi de numero Indorum. Al-Khwārizmī, rendered as (Latin) Algoritmi, led to the term "algorithm".

Some of his work was based on Persian and Babylonian astronomy, 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 of Ptolemy but with improved values for the Mediterranean Sea, Asia, and Africa.[citation needed]

He also wrote on mechanical devices like the astrolabe[43] and sundial. 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.[44] 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.[citation needed]

Algebra

Main article: The Compendious Book on Calculation by Completion and Balancing
Further information: Latin translations of the 12th century, Mathematics in medieval Islam, and Science in the medieval Islamic world
 
 
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.

The Compendious Book on Calculation by Completion and Balancing (Arabic: الكتاب المختصر في حساب الجبر والمقابلة al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala) is a mathematical book written approximately 820 CE. The book was written with the encouragement of Caliph al-Ma'mun as a popular work on calculation and is replete with examples and applications to a wide range of problems in trade, surveying and legal inheritance.[45] The term "algebra" is derived from the name of one of the basic operations with equations (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 also 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.[46]

It provided an exhaustive account of solving polynomial equations up to the second degree,[47] 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.[48]

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 (ax2 = bx)
  • squares equal number (ax2 = c)
  • roots equal number (bx = c)
  • squares and roots equal number (ax2 + bx = c)
  • squares and number equal roots (ax2 + c = bx)
  • roots and number equal squares (bx + c = ax2)

by dividing out the coefficient of the square and using the two operations al-jabr (Arabic: الجبر "restoring" or "completion") and al-muqābala ("balancing"). 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, x2 = 40x − 4x2 is reduced to 5x2 = 40x. Al-muqābala is the process of bringing quantities of the same type to the same side of the equation. For example, x2 + 14 = x + 5 is reduced to x2 + 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 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)

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.[45]

In modern notation this process, with x the "thing" (شيء shayʾ) or "root", is given by the steps,

 
 
 

Let the roots of the equation be x = p and x = q. Then  ,   and

 

So a root is given by

 

Several authors have also published texts under the name of Kitāb al-jabr wal-muqābala, including Abū Ḥanīfa Dīnawarī, Abū Kāmil Shujāʿ ibn Aslam, Abū Muḥammad al-'Adlī, Abū Yūsuf al-Miṣṣīṣī, 'Abd al-Hamīd ibn Turk, Sind ibn 'Alī, Sahl ibn Bišr, and Sharaf al-Dīn al-Ṭūsī.

S. Gandz has described Al-Khwarizmi as the father of Algebra :

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.[49]

Victor J. Katz adds :

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.[50]

J.J. O'Conner and E.F. Robertson wrote in the MacTutor History of Mathematics archive:

Perhaps one of the most significant advances made by 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 numbers, irrational numbers, 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.[51]

R. Rashed and Angela Armstrong write:

Al-Khwarizmi's text can be seen to be distinct not only from the Babylonian tablets, but also from Diophantus' Arithmetica. It no longer concerns a series of problems to be solved, but an 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.[52]

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''.[53] Regarding the dissimilarity and significance of Al-Khwarizmi's algebraic work from that of Indian Mathematician Brahmagupta, Carl Benjamin Boyer wrote:

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.[54]

 
Page from a Latin translation, beginning with "Dixit algorizmi"

Arithmetic

 
Algorists vs. abacists, depicted in a sketch from 1508 CE

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'[note 2]), and perhaps a more elementary text, kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī ('Addition and subtraction in Indian arithmetic').[56][57] These texts described algorithms on decimal numbers (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.[58]

As part of 12th century wave of Arabic science flowing into Europe via translations, these texts proved to be revolutionary in Europe.[59] Al-Khwarizmi's Latinized name, Algorismus, turned into the name of method used for computations, and survives in the modern term "algorithm". It gradually replaced the previous abacus-based methods used in Europe.[60]

Four Latin texts providing adaptions of Al-Khwarizmi's methods have survived, even though none of them is believed to be a literal translation:[56]

  • Dixit Algorizmi (published in 1857 under the title Algoritmi de Numero Indorum[61])[62]
  • 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 also translated the astronomical tables in 1126. It is perhaps the closest to Al-Khwarizmi's own writings.[62]

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 Latinized forms of al-Khwārizmī's name, Algoritmi and Algorismi, respectively.

Astronomy

Further information: Astronomy in the medieval Islamic world
 
Page from Corpus Christi College MS 283, a Latin translation of al-Khwārizmī's Zīj

Al-Khwārizmī's Zīj al-Sindhind[38] (Arabic: زيج السند هند, "astronomical tables of Siddhanta"[63]) 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 Zijes based on the Indian astronomical methods known as the sindhind.[64] 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" (Brahmasphutasiddhanta) of Brahmagupta.[65]

The work contains tables for the movements of the sun, the moon and the five planets 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.

The original Arabic version (written c. 820) is lost, but a version by the Spanish astronomer Maslamah Ibn Ahmad al-Majriti (c. 1000) has survived in a Latin translation, presumably by Adelard of Bath (26 January 1126).[66] 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).

Trigonometry

Al-Khwārizmī's Zīj al-Sindhind also contained tables for the trigonometric functions of sines and cosine.[64] A related treatise on spherical trigonometry is also attributed to him.[51]

Al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents.[67][68]

Geography

 
Hubert Daunicht'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, Martellus and Behaim. The general shape of the coastline is the same between Taprobane and Cattigara. The 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.
 
A 15th-century version of Ptolemy's Geography for comparison

Al-Khwārizmī's third major work is his Kitāb Ṣūrat al-Arḍ (Arabic: كتاب صورة الأرض, "Book of the Description of the Earth"),[69] also known as his Geography, which was finished in 833. It is a major reworking of Ptolemy's second-century Geography, consisting of a list of 2402 coordinates of cities and other geographical features following a general introduction.[70]

There is only one surviving copy of Kitāb Ṣūrat al-Arḍ, which is kept at the Strasbourg University Library. A Latin translation is kept at the Biblioteca Nacional de España in Madrid.[citation needed] 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[dubious – discuss] points out, this excellent 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 itself; 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 deduces 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 then does the same for the rivers and towns.[71]

Al-Khwārizmī corrected Ptolemy's gross overestimate for the length of the Mediterranean Sea[72] 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 "also depicted the Atlantic and Indian Oceans as open bodies of water, not land-locked seas as Ptolemy had done."[73] 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.[72]

Jewish calendar

Al-Khwārizmī wrote several other works including a treatise on the Hebrew calendar, titled Risāla fi istikhrāj ta'rīkh al-yahūd (Arabic: 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 Abū Rayḥān al-Bīrūnī and Maimonides.[38]

Other works

Ibn al-Nadim's Kitāb al-Fihrist, an index of Arabic books, mentions al-Khwārizmī's Kitāb al-Taʾrīkh (Arabic: 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.[74]

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 Kitāb ar-Rukhāma(t) (Arabic: كتاب الرخامة). 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 (Ma'rifat sa'at al-mashriq fī kull balad) and the determination of the azimuth from a height (Ma'rifat al-samt min qibal al-irtifā').

He also wrote two books on using and constructing astrolabes.

Honors

 
A Soviet postage stamp (issued 6 September 1983) commemorating al-Khwārizmī's (approximate) 1200th birthday
 
Wikiquote has quotations related to al-Khwārizmī.
 
Wikimedia Commons has media related to Muhammad ibn Musa al-Khwarizmi.
  • Al-Khwarizmi (crater) — A crater on the far side of the Moon → El-Baz, Farouk (1973). "Al-Khwarizmi: A New-Found Basin on the Lunar Far Side". Science. 180 (4091): 1173–1176. Bibcode:1973Sci...180.1173E. doi:10.1126/science.180.4091.1173. JSTOR 1736378. PMID 17743602. S2CID 10623582. NASA Portal: Apollo 11, Photography Index.
  • 13498 Al Chwarizmi — Main-belt Asteroid, Discovered 1986 Aug 6 by E. W. Elst and V. G. Ivanova at Smolyan.
  • 11156 Al-Khwarismi — Main-belt Asteroid, Discovered 1997 Dec 31 by P. G. Comba at Prescott.

Notes

  1. ^ There is some confusion in the literature on whether al-Khwārizmī's full name is ابو عبد الله محمد بن موسى الخوارزمي Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī or ابو جعفر محمد بن موسی الخوارزمی 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."[4] 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 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.[5]
  2. ^ 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.[55]

References

  1. ^ O'Connor, John J.; Robertson, Edmund F., "Abū Kāmil Shujā' ibn Aslam" 11 December 2013 at the Wayback Machine, MacTutor History of Mathematics archive, University of St Andrews.
  2. ^ Toomer, Gerald J. (1970–1980). "al-Khuwārizmī, Abu Ja'far Muḥammad ibn Mūsā". In Gillispie, Charles Coulston (ed.). Dictionary of Scientific Biography. Vol. VII. pp. 358–365. ISBN 978-0-684-16966-8.
  3. ^ Vernet, Juan (1960–2005). "Al-Khwārizmī". In Gibb, H. A. R.; Kramers, J. H.; Lévi-Provençal, E.; Schacht, J. (eds.). The Encyclopaedia of Islam. Vol. IV (2nd ed.). Leiden: Brill. pp. 1070–1071. OCLC 399624.
  4. ^ Ibn Khaldūn, The Muqaddimah: An introduction to history 17 September 2016 at the Wayback Machine, Translated from the Arabic by Franz Rosenthal, New York: Princeton (1958), Chapter VI:19.
  5. ^ Knuth, Donald (1997), "Basic Concepts", The Art of Computer Programming, vol. 1 (3rd ed.), Addison-Wesley, p. 1, ISBN 978-0-201-89683-1
  6. ^ a b Maher, P. (1998), "From Al-Jabr to Algebra", Mathematics in School, 27(4), 14–15.
  7. ^ Oaks, J. (2009), "Polynomials and Equations in Arabic Algebra", Archive for History of Exact Sciences, 63(2), 169–203.
  8. ^ (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" – that is, the cancellation of like terms on opposite sides of the equation."
  9. ^ Corbin, Henry (1998). The Voyage and the Messenger: Iran and Philosophy. North Atlantic Books. p. 44. ISBN 978-1-55643-269-9. from the original on 28 March 2023. Retrieved 19 October 2020.
  10. ^ Boyer, Carl B., 1985. A History of Mathematics, p. 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..."
  11. ^ S Gandz, 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."
  12. ^ Katz, Victor J. (PDF). VICTOR J.KATZ, University of the District of Columbia Washington DC, USA: 190. Archived from the original (PDF) on 27 March 2019. Retrieved 7 October 2017 – via University of the District of Columbia Washington DC, USA. 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.
  13. ^ Esposito, John L. (6 April 2000). The Oxford History of Islam. Oxford University Press. p. 188. ISBN 978-0-19-988041-6. from the original on 28 March 2023. Retrieved 29 September 2020. Al-Khwarizmi is often considered the founder of algebra, and his name gave rise to the term algorithm.
  14. ^ Brentjes, Sonja (1 June 2007). "Algebra". Encyclopaedia of Islam, THREE. from the original on 22 December 2019. Retrieved 5 June 2019.
  15. ^ Daffa 1977
  16. ^ Clegg, Brian (1 October 2019). Scientifica Historica: How the world's great science books chart the history of knowledge. Ivy Press. p. 61. ISBN 978-1-78240-879-6. from the original on 28 March 2023. Retrieved 30 December 2021.
  17. ^ Knuth, Donald (1979). (PDF). Springer-Verlag. ISBN 978-0-387-11157-5. Archived from the original (PDF) on 7 November 2006.
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  19. ^ Struik 1987, p. 93
  20. ^ Philip Khuri Hitti (2002). . Palgrave Macmillan. p. 379. ISBN 978-1-137-03982-8. Archived from the original on 20 December 2019.
  21. ^ Fred James Hill, Nicholas Awde (2003). A History of the Islamic World. Hippocrene Books. p. 55. ISBN 978-0-7818-1015-9. "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
  22. ^ Shawn Overbay; Jimmy Schorer; Heather Conger. "Al-Khwarizmi". University of Kentucky. from the original on 12 December 2013.
  23. ^ "Islam Spain and the history of technology". www.sjsu.edu. from the original on 11 October 2018. Retrieved 24 January 2018.
  24. ^ Bartel Leenert van der Waerden (1985). A History of Algebra: From al–Khwarizmi to Emmy Noether. Berlin: Springer-Verlag.
  25. ^ Arndt 1983, p. 669
  26. ^ a b Saliba, George (September 1998). "Science and medicine". Iranian Studies. 31 (3–4): 681–690. doi:10.1080/00210869808701940. 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.
  27. ^ Oaks, Jeffrey A. (2014). "Khwārizmī". In Kalin, Ibrahim (ed.). The Oxford Encyclopedia of Philosophy, Science, and Technology in Islam. Vol. 1. Oxford: Oxford University Press. pp. 451–459. ISBN 978-0-19-981257-8. from the original on 30 January 2022. Retrieved 6 September 2021.
    "Ibn al-Nadīm and Ibn al-Qifṭī relate that al-Khwārizmī's family came from Khwārizm, the region south of the Aral sea."
    Also → al-Nadīm, Abu'l-Faraj (1871–1872). Kitāb al-Fihrist, ed. Gustav Flügel, Leipzig: Vogel, p. 274. al-Qifṭī, Jamāl al-Dīn (1903). Taʾrīkh al-Hukamā, eds. August Müller & Julius Lippert, Leipzig: Theodor Weicher, p. 286.
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  30. ^ 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)."
  31. ^ Ben-Menahem, Ari (2009). Historical Encyclopedia of Natural and Mathematical Sciences (1st ed.). Berlin: Springer. pp. 942–943. ISBN 978-3-540-68831-0. Persian mathematician Al-Khowarizmi
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  37. ^ David A. King (7 March 2018). Astronomy in the Service of Islam. Al-Furqān Islamic Heritage Foundation – Centre for the Study of Islamic Manuscripts. Event occurs at 20:51. from the original on 1 December 2021. Retrieved 26 November 2021. 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.
  38. ^ a b c Toomer 1990
  39. ^ Bosworth, C. E., ed. (1987). The History of al-Ṭabarī, Volume XXXII: The Reunification of the ʿAbbāsid Caliphate: The Caliphate of al-Maʾmūn, A.D. 813–33/A.H. 198–213. SUNY Series in Near Eastern Studies. Albany, New York: State University of New York Press. p. 158. ISBN 978-0-88706-058-8.
  40. ^ Golden, Peter; Ben-Shammai, Haggai; Roná-Tas, András (13 August 2007). The World of the Khazars: New Perspectives. Selected Papers from the Jerusalem 1999 International Khazar Colloquium. BRILL. p. 376. ISBN 978-90-474-2145-0.
  41. ^ Dunlop 1943
  42. ^ Yahya Tabesh; Shima Salehi. "Mathematics Education in Iran From Ancient to Modern" (PDF). Sharif University of Technology. (PDF) from the original on 16 April 2018. Retrieved 16 April 2018.
  43. ^ Joseph Frank, al-Khwarizmi über das Astrolab, 1922.
  44. ^ "al-Khwarizmi". Encyclopædia Britannica. from the original on 5 January 2008. Retrieved 30 May 2008.
  45. ^ a b Rosen, Frederic. "The Compendious Book on Calculation by Completion and Balancing, al-Khwārizmī". 1831 English Translation. from the original on 16 July 2011. Retrieved 14 September 2009.
  46. ^ Karpinski, L.C. (1912). "History of Mathematics in the Recent Edition of the Encyclopædia Britannica". Science. 35 (888): 29–31. Bibcode:1912Sci....35...29K. doi:10.1126/science.35.888.29. PMID 17752897. from the original on 30 October 2020. Retrieved 29 September 2020.
  47. ^ Boyer 1991, p. 228: "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."
  48. ^ (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" — that is, the cancellation of like terms on opposite sides of the equation."
  49. ^ S Gandz, The sources of al-Khwarizmi's algebra, Osiris, i (1936), 263–277
  50. ^ Katz, Victor J. (PDF). VICTOR J.KATZ, University of the District of Columbia Washington DC, USA: 190. Archived from the original (PDF) on 27 March 2019. Retrieved 7 October 2017 – via University of the District of Columbia Washington DC, USA.
  51. ^ a b O'Connor, John J.; Robertson, Edmund F., "Abu Ja'far Muhammad ibn Musa Al-Khwarizmi", MacTutor History of Mathematics Archive, University of St Andrews
  52. ^ Rashed, R.; Armstrong, Angela (1994). The Development of Arabic Mathematics. Springer. pp. 11–12. ISBN 978-0-7923-2565-9. OCLC 29181926.
  53. ^ Florian Cajori (1919). A History of Mathematics. Macmillan. p. 103. 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.
  54. ^ Carl Benjamin Boyer (1968). A History of Mathematics. p. 252.
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  58. ^ Van Brummelen, Glen (2017), "Arithmetic", in Thomas F. Glick (ed.), Routledge Revivals: Medieval Science, Technology and Medicine (2006): An Encyclopedia, Taylor & Francis, p. 46, ISBN 978-1-351-67617-5, from the original on 28 March 2023, retrieved 5 May 2019
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  60. ^ Van Brummelen, Glen (2017), "Arithmetic", in Thomas F. Glick (ed.), Routledge Revivals: Medieval Science, Technology and Medicine (2006): An Encyclopedia, Taylor & Francis, pp. 46–47, ISBN 978-1-351-67617-5, from the original on 28 March 2023, retrieved 5 May 2019
  61. ^ "Algoritmi de numero Indorum", Trattati D'Aritmetica, Rome: Tipografia delle Scienze Fisiche e Matematiche, 1857, pp. 1–, from the original on 28 March 2023, retrieved 6 May 2019
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  63. ^ Thurston, Hugh (1996), Early Astronomy, Springer Science & Business Media, pp. 204–, ISBN 978-0-387-94822-5
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  66. ^ Kennedy 1956, p. 128
  67. ^ Jacques Sesiano, "Islamic mathematics", p. 157, in Selin, Helaine; D'Ambrosio, Ubiratan, eds. (2000). Mathematics Across Cultures: The History of Non-western Mathematics. Springer Science+Business Media. ISBN 978-1-4020-0260-1.
  68. ^ "trigonometry". Encyclopædia Britannica. from the original on 6 July 2008. Retrieved 21 July 2008.
  69. ^ 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", 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".
  70. ^ . GAP computer algebra system. Archived from the original on 24 May 2008. Retrieved 30 May 2008.
  71. ^ Daunicht.
  72. ^ a b Edward S. Kennedy, Mathematical Geography, p. 188, in (Rashed & Morelon 1996, pp. 185–201)
  73. ^ Covington, Richard (2007). . Saudi Aramco World, May–June 2007: 17–21. Archived from the original on 12 May 2008. Retrieved 6 July 2008.
  74. ^ LJ Delaporte (1910). Chronographie de Mar Elie bar Sinaya. Paris. p. xiii.{{cite book}}: CS1 maint: location missing publisher (link)

Sources

  • Arndt, A. B. (December 1983). "Al-Khwarizmi". The Mathematics Teacher. 76 (9): 668–670. doi:10.5951/MT.76.9.0668. JSTOR 27963784.
  • Burnett, Charles (2017), "Arabic Numerals", in Thomas F. Glick (ed.), Routledge Revivals: Medieval Science, Technology and Medicine (2006): An Encyclopedia, Taylor & Francis, ISBN 978-1-351-67617-5, from the original on 28 March 2023, retrieved 5 May 2019
  • Dunlop, Douglas Morton (1943). "Muḥammad b. Mūsā al-Khwārizmī". The Journal of the Royal Asiatic Society of Great Britain and Ireland. 2 (3–4): 248–250. doi:10.1017/S0035869X00098464. JSTOR 25221920. S2CID 161841351. from the original on 25 June 2021. Retrieved 24 June 2021.
  • Kennedy, E. S. (1956). "A Survey of Islamic Astronomical Tables". Transactions of the American Philosophical Society. 46 (2): 123–177. doi:10.2307/1005726. hdl:2027/mdp.39076006359272. JSTOR 1005726. from the original on 4 June 2021. Retrieved 24 June 2021.

Further reading

Specific references

Biographical

  • Brentjes, Sonja (2007). "Khwārizmī: Muḥammad ibn Mūsā al‐Khwārizmī 6 July 2011 at the Wayback Machine" in Thomas Hockey et al.(eds.). The Biographical Encyclopedia of Astronomers, Springer Reference. New York: Springer, 2007, pp. 631–633. (PDF version 14 January 2012 at the Wayback Machine)
  • Hogendijk, Jan P., Muhammad ibn Musa (Al-)Khwarizmi (c. 780–850 CE) 3 February 2018 at the Wayback Machine – bibliography of his works, manuscripts, editions and translations.
  • O'Connor, John J.; Robertson, Edmund F., "Abu Ja'far Muhammad ibn Musa Al-Khwarizmi", MacTutor History of Mathematics Archive, University of St Andrews
  • Sezgin, Fuat (1974). Geschichte des arabischen Schrifttums, Band V: Mathematik. Bis ca. 430 H. Leiden: Brill. pp. 228–241.
  • Sezgin, Fuat (1978). Geschichte des arabischen Schrifttums, Band VI: Astronomie. Bis ca. 430 H. Leiden: Brill. pp. 140–143. Bibcode:1978gasb.book.....S.
  • Sezgin, Fuat (1979). Geschichte des arabischen Schrifttums, Band VII: Astrologie – Meteorlogie und Verwanndtes Bis ca. 430 H. Leiden: Brill. pp. 128–129.
  • Sezgin, F., ed., Islamic Mathematics and Astronomy, Frankfurt: Institut für Geschichte der arabisch-islamischen Wissenschaften, 1997–99.

Algebra

  • Gandz, Solomon (November 1926). "The Origin of the Term "Algebra". The American Mathematical Monthly. 33 (9): 437–440. doi:10.2307/2299605. JSTOR 2299605. from the original on 25 June 2021. Retrieved 24 June 2021.
  • Gandz, Solomon (1936). "The Sources of al-Khowārizmī's Algebra". Osiris. 1 (1): 263–277. doi:10.1086/368426. JSTOR 301610. S2CID 60770737. from the original on 25 June 2021. Retrieved 24 June 2021.
  • Gandz, Solomon (1938). "The Algebra of Inheritance: A Rehabilitation of Al-Khuwārizmī". Osiris. 5 (5): 319–391. doi:10.1086/368492. JSTOR 301569. S2CID 143683763. from the original on 25 June 2021. Retrieved 24 June 2021.
  • Hughes, Barnabas (1986). "Gerard of Cremona's Translation of al-Khwārizmī's al-Jabr, A Critical Edition". Mediaeval Studies. 48: 211–263. doi:10.1484/J.MS.2.306339.
  • Barnabas Hughes. 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.
  • Karpinski, L.C. (1915). Robert of Chester's Latin Translation of the Algebra of Al-Khowarizmi: With an Introduction, Critical Notes and an English Version. The Macmillan Company. from the original on 24 September 2020. Retrieved 21 May 2020.
  • Rosen, Fredrick (1831). The Algebra of Mohammed Ben Musa. London.
  • Ruska, Julius (1917). "Zur ältesten arabischen Algebra und Rechenkunst". Sitzungsberichte der Heidelberger Akademie der Wissenschaften. Philologisch-historische Klasse. 2: 1–125.

Arithmetic

  • Folkerts, Menso (1997). Die älteste lateinische Schrift über das indische Rechnen nach al-Ḫwārizmī. München: Bayerische Akademie der Wissenschaften. ISBN 978-3-7696-0108-4. from the original on 24 June 2021. Retrieved 24 June 2021. (This is a new edition of the complete medieval Latin translation of the Arithmetic of al-Khwarizmi, previous editions are all incomplete. This work is lost in Arabic).
  • Vogel, Kurt (1963). Mohammed ibn Musa Alchwarizmi's Algorismus; das früheste Lehrbuch zum Rechnen mit indischen Ziffern. Nach der einzigen (lateinischen) Handschrift (Cambridge Un. Lib. Ms. Ii. 6.5) in Faksimile mit Transkription und Kommentar. Milliaria.3. Aalen, O. Zeller. from the original on 25 June 2021. Retrieved 24 June 2021.

Astronomy

  • Goldstein, B.R. (1968). Commentary on the Astronomical Tables of Al-Khwarizmi: By Ibn Al-Muthanna. Yale University Press. ISBN 978-0-300-00498-4.
  • Hogendijk, Jan P. (1991). "Al-Khwārizmī's Table of the "Sine of the Hours" and the Underlying Sine Table". Historia Scientiarum. 42: 1–12. from the original on 7 May 2021. Retrieved 24 June 2021. (Hogendijk's homepage. Publication in English, no. 25).
  • King, David A. (1983). Al-Khwārizmī and New Trends in Mathematical Astronomy in the Ninth Century. New York University: Hagop Kevorkian Center for Near Eastern Studies: Occasional Papers on the Near East 2. from the original on 25 June 2021. Retrieved 24 June 2021. (Description and analysis of seven recently discovered minor works related to al-Khwarizmi).
  • Neugebauer, Otto (1962). The Astronomical Tables of al-Khwarizmi.
  • Rosenfeld, Boris A. (1993). "'Geometric trigonometry' in treatises of al-Khwārizmī, al-Māhānī and Ibn al-Haytham". In Folkerts, Menso; Hogendijk, Jan P. (eds.). Vestigia Mathematica: Studies in Medieval and Early Modern Mathematics in Honour of H.L.L. Busard. Leiden: Brill. pp. 305–308. ISBN 978-90-5183-536-6.
  • Suter, Heinrich. [Ed.]: Die astronomischen Tafeln des Muhammed ibn Mûsâ al-Khwârizmî in der Bearbeitung des Maslama ibn Ahmed al-Madjrîtî und der latein. Übersetzung des Athelhard von Bath auf Grund der Vorarbeiten von A. Bjørnbo und R. Besthorn in Kopenhagen. Hrsg. und komm. Kopenhagen 1914. 288 pp. Repr. 1997 (Islamic Mathematics and Astronomy. 7). ISBN 3-8298-4008-X.
  • Van Dalen, Benno (1996). "al-Khwârizmî's Astronomical Tables Revisited: Analysis of the Equation of Time". In Casulleras, Josep; Samsó, Julio (eds.). From Baghdad to Barcelona, Studies on the Islamic Exact Sciences in Honour of Prof. Juan Vernet. Barcelona: Instituto Millás Vallicrosa de Historia de la Ciencia Arabe. pp. 195–252. from the original on 24 June 2021. Retrieved 24 June 2021. (Van Dalen's homepage. List of Publications, Articles – no. 5).

Spherical trigonometry

  • B.A. Rozenfeld. "Al-Khwarizmi's spherical trigonometry" (Russian), Istor.-Mat. Issled. 32–33 (1990), 325–339.

Jewish calendar

Geography

  • Daunicht, Hubert (1968–1970). Der Osten nach der Erdkarte al-Ḫuwārizmīs. Beiträge zur historischen Geographie und Geschichte Asiens. Bd 1: Rekonstruktion der Karte, Interpretation der Karte: Südasien; Teil 2: Die ost- und südostasiatische Inselwelt und die Meere; Teil 3: Der Süden des festländischen Ostasiens; Teil 4, 1 u. 2: Der Norden des festländischen Ostasiens und Nord- und Mittelasien. Diss.--Bonn: Bonner Orientalistische Studien. N. S. Bd 19. 19a—d. JSTOR 43370513. from the original on 25 June 2021. Retrieved 24 June 2021.
  • Mžik, Hans von (1915). "Ptolemaeus und die Karten der arabischen Geographen". Mitteil. D. K. K. Geogr. Ges. In Wien. 58: 152.
  • Mžik, Hans von (1916). "Afrika nach der arabischen Bearbeitung der γεωγραφικὴ ὑφήγησις des Cl. Ptolomeaus von Muh. ibn Mūsa al-Hwarizmi". Denkschriften D. Akad. D. Wissen. In Wien, Phil.-hist. Kl. 59.
  • Mžik, Hans von (1926). Das Kitāb Ṣūrat al-Arḍ des Abū Ǧa'far Muḥammad ibn Mūsā al-Ḫuwārizmī. Leipzig.{{cite book}}: CS1 maint: location missing publisher (link)
  • Nallino, C.A. (1896), "Al-Ḫuwārizmī e il suo rifacimento della Geografia di Tolemo", Atti della R. Accad. Dei Lincei, Arno 291, Serie V, Memorie, Classe di Sc. Mor., Vol. II, Rome
  • Ruska, Julius (1918). "Neue Bausteine zur Geschichte der arabischen Geographie". Geographische Zeitschrift. 24: 77–81.
  • Spitta, Wilhelm (1879). "Huwârazmî's Auszug aus der Geographie des Ptolemaios". Zeitschrift der Deutschen Morgenländischen Gesellschaft. 33: 294–297. from the original on 24 June 2021. Retrieved 24 June 2021.

General references

For a more extensive bibliography, see History of mathematics, Mathematics in medieval Islam, and Astronomy in medieval Islam.

October 16, 2023
khwarizmi, other, uses, disambiguation, this, article, needs, additional, citations, verification, please, help, improve, this, article, adding, citations, reliable, sources, unsourced, material, challenged, removed, find, sources, news, newspapers, books, sch. For other uses see Al Khwarizmi disambiguation This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Al Khwarizmi news newspapers books scholar JSTOR January 2023 Learn how and when to remove this template message Muḥammad ibn Musa al Khwarizmi note 1 Arabic محمد بن موسى الخوارزمي c 780 c 850 or al Khwarizmi was an Abbasid polymath from Khwarazm who produced vastly influential works in mathematics astronomy and geography Around 820 CE he was appointed as the astronomer and head of the library of the House of Wisdom in Baghdad 6 14 Muḥammad ibn Musa al Khwarizmiمحمد بن موسى الخوارزميWoodcut panel depicting al KhwarizmiBornc 780DiedAfter 847 2 3 aged c 70 Academic workEraIslamic Golden Age Abbasid era Main interestsMathematics astronomy geographyNotable worksThe Compendious Book on Calculation by Completion and BalancingBook of the Description of the Earth Astronomical tables of SiddhantaNotable ideasTreatises on algebra and Hindu Arabic numeral systemInfluencedAbu Kamil 1 Al Khwarizmi s popularizing treatise on algebra The Compendious Book on Calculation by Completion and Balancing c 813 833 CE 7 171 presented the first systematic solution of linear and quadratic equations One of his principal achievements in algebra was his demonstration of how to solve quadratic equations by completing the square for which he provided geometric justifications 6 14 Because he was the first 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 8 he has been described as the father 9 10 11 or founder 12 13 of algebra The term algebra itself comes from the title of his book the word al jabr meaning completion or rejoining 14 His name gave rise to the terms algorism and algorithm 15 16 the Spanish Italian and Portuguese terms algoritmo and the Spanish guarismo 17 and Portuguese algarismo both meaning digit 18 In the 12th century Latin translations of his textbook on arithmetic Algorithmo de Numero Indorum which codified the various Indian numerals introduced the decimal positional number system to the Western world 19 The Compendious Book on Calculation by Completion and Balancing translated into Latin by Robert of Chester in 1145 was used until the sixteenth century as the principal mathematical text book of European universities 20 21 22 23 In addition to his best known works he revised Ptolemy s Geography listing the longitudes and latitudes of various cities and localities 24 9 He further produced a set of astronomical tables and wrote about calendaric works as well as the astrolabe and the sundial 25 He also made important contributions to trigonometry producing accurate sine and cosine tables and the first table of tangents citation needed Contents 1 Life 2 Contributions 2 1 Algebra 2 2 Arithmetic 2 3 Astronomy 2 4 Trigonometry 2 5 Geography 2 6 Jewish calendar 2 7 Other works 3 Honors 4 Notes 5 References 6 Sources 7 Further reading 7 1 Specific references 7 1 1 Biographical 7 1 2 Algebra 7 1 3 Arithmetic 7 1 4 Astronomy 7 1 5 Spherical trigonometry 7 1 6 Jewish calendar 7 1 7 Geography 7 2 General referencesLife nbsp Monument to Muhammad ibn Musa al Khwarizmi at Ciudad Universitaria of MadridFew details of al Khwarizmi 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 26 27 28 Of Persian stock 29 26 30 31 32 his name means the native of Khwarazm a region that was part of Greater Iran 33 and is now part of Turkmenistan and Uzbekistan 34 Muhammad ibn Jarir al Tabari gives his name as Muḥammad ibn Musa al Khwarizmi al Majusi al Quṭrubbulli محمد بن موسى الخوارزمي المجوسـي القطرب ـلي The epithet al Qutrubbulli could indicate he might instead have come from Qutrubbul Qatrabbul 35 near Baghdad However Rashed denies this 36 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 Musa al Khwarizmi and al Majusi al Qutrubbulli and that there are two people al Khwarizmi and al Majusi al Qutrubbulli between whom the letter wa Arabic و for the conjunction and has been omitted in an early copy This would not be worth mentioning if a series of errors concerning the personality of al Khwarizmi occasionally even the origins of his knowledge had not been made Recently 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 affirms his nisba to Qutrubul noting that he was called al Khwarizmi al Qutrubbulli because he was born just outside of Baghdad 37 Regarding al Khwarizmi s religion Toomer writes 38 Another epithet given to him by al Ṭabari al Majusi would seem to indicate that he was an adherent of the old Zoroastrian religion This would still have been possible at that time for a man of Iranian origin but the pious preface to al Khwarizmi s Algebra shows that he was an orthodox Muslim so al Ṭabari s epithet could mean no more than that his forebears and perhaps he in his youth had been Zoroastrians Ibn al Nadim s Kitab al Fihrist includes a short biography on al Khwarizmi together with a list of his books Al Khwarizmi 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 He worked in the House of Wisdom established by the Abbasid Caliph al Ma mun where he studied the sciences and mathematics including the translation of Greek and Sanskrit scientific manuscripts He was also a historian who is frequently cited by the likes of al Tabari and Ibn Abi Tahir 39 During the reign of al Wathiq he is said to have been involved in the first of two embassies to the Khazars 40 Douglas Morton Dunlop suggests that Muḥammad ibn Musa al Khwarizmi might have been the same person as Muḥammad ibn Musa ibn Shakir the eldest of the three Banu Musa 41 Contributions nbsp A page from al Khwarizmi s AlgebraAl Khwarizmi 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 The Compendious Book on Calculation by Completion and Balancing 42 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 It was translated into Latin as Algoritmi de numero Indorum Al Khwarizmi rendered as Latin Algoritmi led to the term algorithm Some of his work was based on Persian and Babylonian astronomy Indian numbers and Greek mathematics Al Khwarizmi 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 of Ptolemy but with improved values for the Mediterranean Sea Asia and Africa citation needed He also wrote on mechanical devices like the astrolabe 43 and sundial 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 44 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 citation needed Algebra Main article The Compendious Book on Calculation by Completion and Balancing Further information Latin translations of the 12th century Mathematics in medieval Islam and Science in the medieval Islamic world nbsp nbsp Left The original Arabic print manuscript of the Book of Algebra by Al Khwarizmi Right A page from The Algebra of Al Khwarizmi by Fredrick Rosen in English The Compendious Book on Calculation by Completion and Balancing Arabic الكتاب المختصر في حساب الجبر والمقابلة al Kitab al mukhtaṣar fi ḥisab al jabr wal muqabala is a mathematical book written approximately 820 CE The book was written with the encouragement of Caliph al Ma mun as a popular work on calculation and is replete with examples and applications to a wide range of problems in trade surveying and legal inheritance 45 The term algebra is derived from the name of one of the basic operations with equations 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 also 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 46 It provided an exhaustive account of solving polynomial equations up to the second degree 47 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 48 Al Khwarizmi 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 ax2 bx squares equal number ax2 c roots equal number bx c squares and roots equal number ax2 bx c squares and number equal roots ax2 c bx roots and number equal squares bx c ax2 by dividing out the coefficient of the square and using the two operations al jabr Arabic الجبر restoring or completion and al muqabala balancing 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 x2 40x 4x2 is reduced to 5x2 40x Al muqabala is the process of bringing quantities of the same type to the same side of the equation For example x2 14 x 5 is reduced to x2 9 x The above discussion uses modern mathematical notation for the types of problems that the book discusses However in al Khwarizmi s day most of this 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 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 45 In modern notation this process with x the thing شيء shayʾ or root is given by the steps 10 x 2 81 x displaystyle 10 x 2 81x nbsp 100 x 2 20 x 81 x displaystyle 100 x 2 20x 81x nbsp x 2 100 101 x displaystyle x 2 100 101x nbsp Let the roots of the equation be x p and x q Then p q 2 50 1 2 displaystyle tfrac p q 2 50 tfrac 1 2 nbsp p q 100 displaystyle pq 100 nbsp and p q 2 p q 2 2 p q 2550 1 4 100 49 1 2 displaystyle frac p q 2 sqrt left frac p q 2 right 2 pq sqrt 2550 tfrac 1 4 100 49 tfrac 1 2 nbsp So a root is given by x 50 1 2 49 1 2 1 displaystyle x 50 tfrac 1 2 49 tfrac 1 2 1 nbsp Several authors have also published texts under the name of Kitab al jabr wal muqabala including Abu Ḥanifa Dinawari Abu Kamil Shujaʿ ibn Aslam Abu Muḥammad al Adli Abu Yusuf al Miṣṣiṣi Abd al Hamid ibn Turk Sind ibn Ali Sahl ibn Bisr and Sharaf al Din al Ṭusi S Gandz has described Al Khwarizmi as the father of Algebra 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 49 Victor J Katz adds 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 50 J J O Conner and E F Robertson wrote in the MacTutor History of Mathematics archive Perhaps one of the most significant advances made by 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 numbers irrational numbers 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 51 R Rashed and Angela Armstrong write Al Khwarizmi s text can be seen to be distinct not only from the Babylonian tablets but also from Diophantus Arithmetica It no longer concerns a series of problems to be solved but an 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 52 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 53 Regarding the dissimilarity and significance of Al Khwarizmi s algebraic work from that of Indian Mathematician Brahmagupta Carl Benjamin Boyer wrote 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 54 nbsp Page from a Latin translation beginning with Dixit algorizmi Arithmetic nbsp Algorists vs abacists depicted in a sketch from 1508 CEAl Khwarizmi 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 kitab al ḥisab al hindi Book of Indian computation note 2 and perhaps a more elementary text kitab al jam wa l tafriq al ḥisab al hindi Addition and subtraction in Indian arithmetic 56 57 These texts described algorithms on decimal numbers 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 58 As part of 12th century wave of Arabic science flowing into Europe via translations these texts proved to be revolutionary in Europe 59 Al Khwarizmi s Latinized name Algorismus turned into the name of method used for computations and survives in the modern term algorithm It gradually replaced the previous abacus based methods used in Europe 60 Four Latin texts providing adaptions of Al Khwarizmi s methods have survived even though none of them is believed to be a literal translation 56 Dixit Algorizmi published in 1857 under the title Algoritmi de Numero Indorum 61 62 Liber Alchoarismi de Practica Arismetice Liber Ysagogarum Alchorismi Liber PulverisDixit 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 also translated the astronomical tables in 1126 It is perhaps the closest to Al Khwarizmi s own writings 62 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 Khwarizmi Both algorithm and algorism are derived from the Latinized forms of al Khwarizmi s name Algoritmi and Algorismi respectively Astronomy Further information Astronomy in the medieval Islamic world nbsp Page from Corpus Christi College MS 283 a Latin translation of al Khwarizmi s ZijAl Khwarizmi s Zij al Sindhind 38 Arabic زيج السند هند astronomical tables of Siddhanta 63 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 Zijes based on the Indian astronomical methods known as the sindhind 64 The word Sindhind is a corruption of the Sanskrit Siddhanta 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 Brahmasphutasiddhanta of Brahmagupta 65 The work contains tables for the movements of the sun the moon and the five planets 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 The original Arabic version written c 820 is lost but a version by the Spanish astronomer Maslamah Ibn Ahmad al Majriti c 1000 has survived in a Latin translation presumably by Adelard of Bath 26 January 1126 66 The four surviving manuscripts of the Latin translation are kept at the Bibliotheque publique Chartres the Bibliotheque Mazarine Paris the Biblioteca Nacional Madrid and the Bodleian Library Oxford Trigonometry Al Khwarizmi s Zij al Sindhind also contained tables for the trigonometric functions of sines and cosine 64 A related treatise on spherical trigonometry is also attributed to him 51 Al Khwarizmi produced accurate sine and cosine tables and the first table of tangents 67 68 Geography nbsp Hubert Daunicht s reconstruction of the section of al Khwarizmi s world map concerning the Indian Ocean The majority of the placenames used by al Khwarizmi match those of Ptolemy Martellus and Behaim The general shape of the coastline is the same between Taprobane and Cattigara The 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 Khwarizmi s map although is clear and precise on the Martellus map and on the later Behaim version nbsp A 15th century version of Ptolemy s Geography for comparisonAl Khwarizmi s third major work is his Kitab Ṣurat al Arḍ Arabic كتاب صورة الأرض Book of the Description of the Earth 69 also known as his Geography which was finished in 833 It is a major reworking of Ptolemy s second century Geography consisting of a list of 2402 coordinates of cities and other geographical features following a general introduction 70 There is only one surviving copy of Kitab Ṣurat al Arḍ which is kept at the Strasbourg University Library A Latin translation is kept at the Biblioteca Nacional de Espana in Madrid citation needed 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 dubious discuss points out this excellent 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 itself 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 deduces 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 then does the same for the rivers and towns 71 Al Khwarizmi corrected Ptolemy s gross overestimate for the length of the Mediterranean Sea 72 from the Canary Islands to the eastern shores of the Mediterranean Ptolemy overestimated it at 63 degrees of longitude while al Khwarizmi almost correctly estimated it at nearly 50 degrees of longitude He also depicted the Atlantic and Indian Oceans as open bodies of water not land locked seas as Ptolemy had done 73 Al Khwarizmi 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 Khwarizmi s prime meridian 72 Jewish calendar Al Khwarizmi wrote several other works including a treatise on the Hebrew calendar titled Risala fi istikhraj ta rikh al yahud Arabic 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 Abu Rayḥan al Biruni and Maimonides 38 Other works Ibn al Nadim s Kitab al Fihrist an index of Arabic books mentions al Khwarizmi s Kitab al Taʾrikh Arabic 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 74 Several Arabic manuscripts in Berlin Istanbul Tashkent Cairo and Paris contain further material that surely or with some probability comes from al Khwarizmi The Istanbul manuscript contains a paper on sundials the Fihrist credits al Khwarizmi with Kitab ar Rukhama t Arabic كتاب الرخامة 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 Ma rifat sa at al mashriq fi kull balad and the determination of the azimuth from a height Ma rifat al samt min qibal al irtifa He also wrote two books on using and constructing astrolabes Honors nbsp A Soviet postage stamp issued 6 September 1983 commemorating al Khwarizmi s approximate 1200th birthday nbsp Wikiquote has quotations related to al Khwarizmi nbsp Wikimedia Commons has media related to Muhammad ibn Musa al Khwarizmi Al Khwarizmi crater A crater on the far side of the Moon El Baz Farouk 1973 Al Khwarizmi A New Found Basin on the Lunar Far Side Science 180 4091 1173 1176 Bibcode 1973Sci 180 1173E doi 10 1126 science 180 4091 1173 JSTOR 1736378 PMID 17743602 S2CID 10623582 NASA Portal Apollo 11 Photography Index 13498 Al Chwarizmi Main belt Asteroid Discovered 1986 Aug 6 by E W Elst and V G Ivanova at Smolyan 11156 Al Khwarismi Main belt Asteroid Discovered 1997 Dec 31 by P G Comba at Prescott Notes There is some confusion in the literature on whether al Khwarizmi s full name is ابو عبد الله محمد بن موسى الخوارزمي Abu ʿAbdallah Muḥammad ibn Musa al Khwarizmi or ابو جعفر محمد بن موسی الخوارزمی Abu Ja far Muḥammad ibn Musa al Khwarizmi 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 4 In the introduction to his critical commentary on Robert of Chester s Latin translation of al Khwarizmi s Algebra L C Karpinski notes that Abu Ja far Muḥammad ibn Musa refers to the eldest of the Banu Musa brothers Karpinski notes in his review on Ruska 1917 that in Ruska 1918 Ruska here inadvertently speaks of the author as Abu Ga far M b M instead of Abu Abdallah M b M Donald Knuth writes it as Abu Abd Allah Muḥammad ibn Musa al Khwarizmi and quotes it as meaning literally Father of Abdullah Mohammed son of Moses native of Khwarizm citing previous work by Heinz Zemanek 5 Some scholars translate the title al ḥisab al hindi as computation with Hindu numerals but Arabic Hindi 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 55 References O Connor John J Robertson Edmund F Abu Kamil Shuja ibn Aslam Archived 11 December 2013 at the Wayback Machine MacTutor History of Mathematics archive University of St Andrews Toomer Gerald J 1970 1980 al Khuwarizmi Abu Ja far Muḥammad ibn Musa In Gillispie Charles Coulston ed Dictionary of Scientific Biography Vol VII pp 358 365 ISBN 978 0 684 16966 8 Vernet Juan 1960 2005 Al Khwarizmi In Gibb H A R Kramers J H Levi Provencal E Schacht J eds The Encyclopaedia of Islam Vol IV 2nd ed Leiden Brill pp 1070 1071 OCLC 399624 Ibn Khaldun The Muqaddimah An introduction to history Archived 17 September 2016 at the Wayback Machine Translated from the Arabic by Franz Rosenthal New York Princeton 1958 Chapter VI 19 Knuth Donald 1997 Basic Concepts The Art of Computer Programming vol 1 3rd ed Addison Wesley p 1 ISBN 978 0 201 89683 1 a b Maher P 1998 From Al Jabr to Algebra Mathematics in School 27 4 14 15 Oaks J 2009 Polynomials and Equations in Arabic Algebra Archive for History of Exact Sciences 63 2 169 203 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 that is the cancellation of like terms on opposite sides of the equation Corbin Henry 1998 The Voyage and the Messenger Iran and Philosophy North Atlantic Books p 44 ISBN 978 1 55643 269 9 Archived from the original on 28 March 2023 Retrieved 19 October 2020 Boyer Carl B 1985 A History of Mathematics p 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 S Gandz 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 Katz Victor J Stages in the History of Algebra with Implications for Teaching PDF VICTOR J KATZ University of the District of Columbia Washington DC USA 190 Archived from the original PDF on 27 March 2019 Retrieved 7 October 2017 via University of the District of Columbia Washington DC USA 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 Esposito John L 6 April 2000 The Oxford History of Islam Oxford University Press p 188 ISBN 978 0 19 988041 6 Archived from the original on 28 March 2023 Retrieved 29 September 2020 Al Khwarizmi is often considered the founder of algebra and his name gave rise to the term algorithm Brentjes Sonja 1 June 2007 Algebra Encyclopaedia of Islam THREE Archived from the original on 22 December 2019 Retrieved 5 June 2019 Daffa 1977 Clegg Brian 1 October 2019 Scientifica Historica How the world s great science books chart the history of knowledge Ivy Press p 61 ISBN 978 1 78240 879 6 Archived from the original on 28 March 2023 Retrieved 30 December 2021 Knuth Donald 1979 Algorithms in Modern Mathematics and Computer Science PDF Springer Verlag ISBN 978 0 387 11157 5 Archived from the original PDF on 7 November 2006 Gandz Solomon 1926 The Origin of the Term Algebra The American Mathematical Monthly 33 9 437 440 doi 10 2307 2299605 ISSN 0002 9890 Struik 1987 p 93 Philip Khuri Hitti 2002 History of the Arabs Palgrave Macmillan p 379 ISBN 978 1 137 03982 8 Archived from the original on 20 December 2019 Fred James Hill Nicholas Awde 2003 A History of the Islamic World Hippocrene Books p 55 ISBN 978 0 7818 1015 9 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 Shawn Overbay Jimmy Schorer Heather Conger Al Khwarizmi University of Kentucky Archived from the original on 12 December 2013 Islam Spain and the history of technology www sjsu edu Archived from the original on 11 October 2018 Retrieved 24 January 2018 Bartel Leenert van der Waerden 1985 A History of Algebra From al Khwarizmi to Emmy Noether Berlin Springer Verlag Arndt 1983 p 669 a b Saliba George September 1998 Science and medicine Iranian Studies 31 3 4 681 690 doi 10 1080 00210869808701940 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 Oaks Jeffrey A 2014 Khwarizmi In Kalin Ibrahim ed The Oxford Encyclopedia of Philosophy Science and Technology in Islam Vol 1 Oxford Oxford University Press pp 451 459 ISBN 978 0 19 981257 8 Archived from the original on 30 January 2022 Retrieved 6 September 2021 Ibn al Nadim and Ibn al Qifṭi relate that al Khwarizmi s family came from Khwarizm the region south of the Aral sea Also al Nadim Abu l Faraj 1871 1872 Kitab al Fihrist ed Gustav Flugel Leipzig Vogel p 274 al Qifṭi Jamal al Din 1903 Taʾrikh al Hukama eds August Muller amp Julius Lippert Leipzig Theodor Weicher p 286 Dodge Bayard ed 1970 The Fihrist of al Nadim A Tenth Century Survey of Islamic Culture vol 2 translated by Dodge New York Columbia University Press Clifford A Pickover 2009 The Math Book From Pythagoras to the 57th Dimension 250 Milestones in the History of Mathematics Sterling Publishing Company Inc p 84 ISBN 978 1 4027 5796 9 Archived from the original on 28 March 2023 Retrieved 19 October 2020 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 Ben Menahem Ari 2009 Historical Encyclopedia of Natural and Mathematical Sciences 1st ed Berlin Springer pp 942 943 ISBN 978 3 540 68831 0 Persian mathematician Al Khowarizmi Wiesner Hanks Merry E Ebrey Patricia Buckley Beck Roger B Davila Jerry Crowston Clare Haru McKay John P 2017 A History of World Societies 11th ed Bedford St Martin s p 419 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 Encycloaedia Iranica online s v CHORASMIA ii In Islamic times Archived 2 September 2021 at the Wayback Machine by Clifford E Bosworth Bosworth Clifford Edmund 1960 2005 Khwarazm In Gibb H A R Kramers J H Levi Provencal E Schacht J eds The Encyclopaedia of Islam Vol IV 2nd ed Leiden Brill pp 1060 1065 OCLC 399624 Iraq After the Muslim Conquest by Michael G Morony ISBN 1 59333 315 3 a 2005 facsimile from the original 1984 book p 145 Archived 27 June 2014 at the Wayback Machine Rashed Roshdi 1988 al Khwarizmi s Concept of Algebra In Zurayq Qusṭanṭin Atiyeh George Nicholas Oweiss Ibrahim M eds Arab Civilization Challenges and Responses Studies in Honor of Constantine K Zurayk SUNY Press p 108 ISBN 978 0 88706 698 6 Archived from the original on 28 March 2023 Retrieved 19 October 2015 David A King 7 March 2018 Astronomy in the Service of Islam Al Furqan Islamic Heritage Foundation Centre for the Study of Islamic Manuscripts Event occurs at 20 51 Archived from the original on 1 December 2021 Retrieved 26 November 2021 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 a b c Toomer 1990 Bosworth C E ed 1987 The History of al Ṭabari Volume XXXII The Reunification of the ʿAbbasid Caliphate The Caliphate of al Maʾmun A D 813 33 A H 198 213 SUNY Series in Near Eastern Studies Albany New York State University of New York Press p 158 ISBN 978 0 88706 058 8 Golden Peter Ben Shammai Haggai Rona Tas Andras 13 August 2007 The World of the Khazars New Perspectives Selected Papers from the Jerusalem 1999 International Khazar Colloquium BRILL p 376 ISBN 978 90 474 2145 0 Dunlop 1943 Yahya Tabesh Shima Salehi Mathematics Education in Iran From Ancient to Modern PDF Sharif University of Technology Archived PDF from the original on 16 April 2018 Retrieved 16 April 2018 Joseph Frank al Khwarizmi uber das Astrolab 1922 al Khwarizmi Encyclopaedia Britannica Archived from the original on 5 January 2008 Retrieved 30 May 2008 a b Rosen Frederic The Compendious Book on Calculation by Completion and Balancing al Khwarizmi 1831 English Translation Archived from the original on 16 July 2011 Retrieved 14 September 2009 Karpinski L C 1912 History of Mathematics in the Recent Edition of the Encyclopaedia Britannica Science 35 888 29 31 Bibcode 1912Sci 35 29K doi 10 1126 science 35 888 29 PMID 17752897 Archived from the original on 30 October 2020 Retrieved 29 September 2020 Boyer 1991 p 228 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 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 that is the cancellation of like terms on opposite sides of the equation S Gandz The sources of al Khwarizmi s algebra Osiris i 1936 263 277 Katz Victor J Stages in the History of Algebra with Implications for Teaching PDF VICTOR J KATZ University of the District of Columbia Washington DC USA 190 Archived from the original PDF on 27 March 2019 Retrieved 7 October 2017 via University of the District of Columbia Washington DC USA a b O Connor John J Robertson Edmund F Abu Ja far Muhammad ibn Musa Al Khwarizmi MacTutor History of Mathematics Archive University of St Andrews Rashed R Armstrong Angela 1994 The Development of Arabic Mathematics Springer pp 11 12 ISBN 978 0 7923 2565 9 OCLC 29181926 Florian Cajori 1919 A History of Mathematics Macmillan p 103 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 Carl Benjamin Boyer 1968 A History of Mathematics p 252 Saidan A S Winter 1966 The Earliest Extant Arabic Arithmetic Kitab al Fusul fi al Hisab al Hindi of Abu al Hasan Ahmad ibn Ibrahim al Uqlidisi Isis The University of Chicago Press 57 4 475 490 doi 10 1086 350163 JSTOR 228518 S2CID 143979243 a b Burnett 2017 p 39 Avari Burjor 2013 Islamic Civilization in South Asia A history of Muslim power and presence in the Indian subcontinent Routledge pp 31 32 ISBN 978 0 415 58061 8 archived from the original on 28 March 2023 retrieved 29 September 2020 Van Brummelen Glen 2017 Arithmetic in Thomas F Glick ed Routledge Revivals Medieval Science Technology and Medicine 2006 An Encyclopedia Taylor amp Francis p 46 ISBN 978 1 351 67617 5 archived from the original on 28 March 2023 retrieved 5 May 2019 Thomas F Glick ed 2017 Al Khwarizmi Routledge Revivals Medieval Science Technology and Medicine 2006 An Encyclopedia Taylor amp Francis ISBN 978 1 351 67617 5 archived from the original on 28 March 2023 retrieved 6 May 2019 Van Brummelen Glen 2017 Arithmetic in Thomas F Glick ed Routledge Revivals Medieval Science Technology and Medicine 2006 An Encyclopedia Taylor amp Francis pp 46 47 ISBN 978 1 351 67617 5 archived from the original on 28 March 2023 retrieved 5 May 2019 Algoritmi de numero Indorum Trattati D Aritmetica Rome Tipografia delle Scienze Fisiche e Matematiche 1857 pp 1 archived from the original on 28 March 2023 retrieved 6 May 2019 a b Crossley John N Henry Alan S 1990 Thus Spake al Khwarizmi A Translation of the Text of Cambridge University Library Ms Ii vi 5 Historia Mathematica 17 2 103 131 doi 10 1016 0315 0860 90 90048 I Thurston Hugh 1996 Early Astronomy Springer Science amp Business Media pp 204 ISBN 978 0 387 94822 5 a b Kennedy 1956 pp 26 29 Waerden Bartel L van der 1985 A History of Algebra From al Khwarizmi to Emmy Noether Berlin Heidelberg Springer Verlag p 10 ISBN 978 3 642 51601 6 Archived from the original on 24 June 2021 Retrieved 22 June 2021 Kennedy 1956 p 128 Jacques Sesiano Islamic mathematics p 157 in Selin Helaine D Ambrosio Ubiratan eds 2000 Mathematics Across Cultures The History of Non western Mathematics Springer Science Business Media ISBN 978 1 4020 0260 1 trigonometry Encyclopaedia Britannica Archived from the original on 6 July 2008 Retrieved 21 July 2008 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 Khwarizmi according to the Geographical Treatise written by Ptolemy the Claudian 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 The history of cartography GAP computer algebra system Archived from the original on 24 May 2008 Retrieved 30 May 2008 Daunicht a b Edward S Kennedy Mathematical Geography p 188 in Rashed amp Morelon 1996 pp 185 201 Covington Richard 2007 The Third Dimension Saudi Aramco World May June 2007 17 21 Archived from the original on 12 May 2008 Retrieved 6 July 2008 LJ Delaporte 1910 Chronographie de Mar Elie bar Sinaya Paris p xiii a href Template Cite book html title Template Cite book cite book a CS1 maint location missing publisher link SourcesArndt A B December 1983 Al Khwarizmi The Mathematics Teacher 76 9 668 670 doi 10 5951 MT 76 9 0668 JSTOR 27963784 Boyer Carl B 1991 The Arabic Hegemony A History of Mathematics Second ed John Wiley amp Sons Inc ISBN 978 0 471 54397 8 Burnett Charles 2017 Arabic Numerals in Thomas F Glick ed Routledge Revivals Medieval Science Technology and Medicine 2006 An Encyclopedia Taylor amp Francis ISBN 978 1 351 67617 5 archived from the original on 28 March 2023 retrieved 5 May 2019Daffa Ali Abdullah al 1977 The Muslim contribution to mathematics London Croom Helm ISBN 978 0 85664 464 1 Dunlop Douglas Morton 1943 Muḥammad b Musa al Khwarizmi The Journal of the Royal Asiatic Society of Great Britain and Ireland 2 3 4 248 250 doi 10 1017 S0035869X00098464 JSTOR 25221920 S2CID 161841351 Archived from the original on 25 June 2021 Retrieved 24 June 2021 Kennedy E S 1956 A Survey of Islamic Astronomical Tables Transactions of the American Philosophical Society 46 2 123 177 doi 10 2307 1005726 hdl 2027 mdp 39076006359272 JSTOR 1005726 Archived from the original on 4 June 2021 Retrieved 24 June 2021 Rashed Roshdi Morelon Regis 1996 Encyclopedia of the History of Arabic Science vol 1 Routledge ISBN 0 415 12410 7Struik Dirk Jan 1987 A Concise History of Mathematics 4th ed Dover Publications ISBN 978 0 486 60255 4 Toomer Gerald 1990 Al Khwarizmi Abu Ja far Muḥammad ibn Musa In Gillispie Charles Coulston ed Dictionary of Scientific Biography Vol 7 New York Charles Scribner s Sons ISBN 978 0 684 16962 0 Archived from the original on 2 July 2016 Retrieved 31 December 2010 Further readingThis further reading section may contain inappropriate or excessive suggestions that may not follow Wikipedia s guidelines Please ensure that only a reasonable number of balanced topical reliable and notable further reading suggestions are given removing less relevant or redundant publications with the same point of view where appropriate Consider utilising appropriate texts as inline sources or creating a separate bibliography article February 2023 Learn how and when to remove this template message Specific references Biographical Brentjes Sonja 2007 Khwarizmi Muḥammad ibn Musa al Khwarizmi Archived 6 July 2011 at the Wayback Machine in Thomas Hockey et al eds The Biographical Encyclopedia of Astronomers Springer Reference New York Springer 2007 pp 631 633 PDF version Archived 14 January 2012 at the Wayback Machine Hogendijk Jan P Muhammad ibn Musa Al Khwarizmi c 780 850 CE Archived 3 February 2018 at the Wayback Machine bibliography of his works manuscripts editions and translations O Connor John J Robertson Edmund F Abu Ja far Muhammad ibn Musa Al Khwarizmi MacTutor History of Mathematics Archive University of St Andrews Sezgin Fuat 1974 Geschichte des arabischen Schrifttums Band V Mathematik Bis ca 430 H Leiden Brill pp 228 241 Sezgin Fuat 1978 Geschichte des arabischen Schrifttums Band VI Astronomie Bis ca 430 H Leiden Brill pp 140 143 Bibcode 1978gasb book S Sezgin Fuat 1979 Geschichte des arabischen Schrifttums Band VII Astrologie Meteorlogie und Verwanndtes Bis ca 430 H Leiden Brill pp 128 129 Sezgin F ed Islamic Mathematics and Astronomy Frankfurt Institut fur Geschichte der arabisch islamischen Wissenschaften 1997 99 Algebra Gandz Solomon November 1926 The Origin of the Term Algebra The American Mathematical Monthly 33 9 437 440 doi 10 2307 2299605 JSTOR 2299605 Archived from the original on 25 June 2021 Retrieved 24 June 2021 Gandz Solomon 1936 The Sources of al Khowarizmi s Algebra Osiris 1 1 263 277 doi 10 1086 368426 JSTOR 301610 S2CID 60770737 Archived from the original on 25 June 2021 Retrieved 24 June 2021 Gandz Solomon 1938 The Algebra of Inheritance A Rehabilitation of Al Khuwarizmi Osiris 5 5 319 391 doi 10 1086 368492 JSTOR 301569 S2CID 143683763 Archived from the original on 25 June 2021 Retrieved 24 June 2021 Hughes Barnabas 1986 Gerard of Cremona s Translation of al Khwarizmi s al Jabr A Critical Edition Mediaeval Studies 48 211 263 doi 10 1484 J MS 2 306339 Barnabas Hughes 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 Karpinski L C 1915 Robert of Chester s Latin Translation of the Algebra of Al Khowarizmi With an Introduction Critical Notes and an English Version The Macmillan Company Archived from the original on 24 September 2020 Retrieved 21 May 2020 Rosen Fredrick 1831 The Algebra of Mohammed Ben Musa London Ruska Julius 1917 Zur altesten arabischen Algebra und Rechenkunst Sitzungsberichte der Heidelberger Akademie der Wissenschaften Philologisch historische Klasse 2 1 125 Arithmetic Folkerts Menso 1997 Die alteste lateinische Schrift uber das indische Rechnen nach al Ḫwarizmi Munchen Bayerische Akademie der Wissenschaften ISBN 978 3 7696 0108 4 Archived from the original on 24 June 2021 Retrieved 24 June 2021 This is a new edition of the complete medieval Latin translation of the Arithmetic of al Khwarizmi previous editions are all incomplete This work is lost in Arabic Vogel Kurt 1963 Mohammed ibn Musa Alchwarizmi s Algorismus das fruheste Lehrbuch zum Rechnen mit indischen Ziffern Nach der einzigen lateinischen Handschrift Cambridge Un Lib Ms Ii 6 5 in Faksimile mit Transkription und Kommentar Milliaria 3 Aalen O Zeller Archived from the original on 25 June 2021 Retrieved 24 June 2021 Astronomy Goldstein B R 1968 Commentary on the Astronomical Tables of Al Khwarizmi By Ibn Al Muthanna Yale University Press ISBN 978 0 300 00498 4 Hogendijk Jan P 1991 Al Khwarizmi s Table of the Sine of the Hours and the Underlying Sine Table Historia Scientiarum 42 1 12 Archived from the original on 7 May 2021 Retrieved 24 June 2021 Hogendijk s homepage Publication in English no 25 King David A 1983 Al Khwarizmi and New Trends in Mathematical Astronomy in the Ninth Century New York University Hagop Kevorkian Center for Near Eastern Studies Occasional Papers on the Near East 2 Archived from the original on 25 June 2021 Retrieved 24 June 2021 Description and analysis of seven recently discovered minor works related to al Khwarizmi Neugebauer Otto 1962 The Astronomical Tables of al Khwarizmi Rosenfeld Boris A 1993 Geometric trigonometry in treatises of al Khwarizmi al Mahani and Ibn al Haytham In Folkerts Menso Hogendijk Jan P eds Vestigia Mathematica Studies in Medieval and Early Modern Mathematics in Honour of H L L Busard Leiden Brill pp 305 308 ISBN 978 90 5183 536 6 Suter Heinrich Ed Die astronomischen Tafeln des Muhammed ibn Musa al Khwarizmi in der Bearbeitung des Maslama ibn Ahmed al Madjriti und der latein Ubersetzung des Athelhard von Bath auf Grund der Vorarbeiten von A Bjornbo und R Besthorn in Kopenhagen Hrsg und komm Kopenhagen 1914 288 pp Repr 1997 Islamic Mathematics and Astronomy 7 ISBN 3 8298 4008 X Van Dalen Benno 1996 al Khwarizmi s Astronomical Tables Revisited Analysis of the Equation of Time In Casulleras Josep Samso Julio eds From Baghdad to Barcelona Studies on the Islamic Exact Sciences in Honour of Prof Juan Vernet Barcelona Instituto Millas Vallicrosa de Historia de la Ciencia Arabe pp 195 252 Archived from the original on 24 June 2021 Retrieved 24 June 2021 Van Dalen s homepage List of Publications Articles no 5 Spherical trigonometry B A Rozenfeld Al Khwarizmi s spherical trigonometry Russian Istor Mat Issled 32 33 1990 325 339 Jewish calendar Kennedy E S 1964 Al Khwarizmi on the Jewish Calendar Scripta Mathematica 27 55 59 Geography Daunicht Hubert 1968 1970 Der Osten nach der Erdkarte al Ḫuwarizmis Beitrage zur historischen Geographie und Geschichte Asiens Bd 1 Rekonstruktion der Karte Interpretation der Karte Sudasien Teil 2 Die ost und sudostasiatische Inselwelt und die Meere Teil 3 Der Suden des festlandischen Ostasiens Teil 4 1 u 2 Der Norden des festlandischen Ostasiens und Nord und Mittelasien Diss Bonn Bonner Orientalistische Studien N S Bd 19 19a d JSTOR 43370513 Archived from the original on 25 June 2021 Retrieved 24 June 2021 Mzik Hans von 1915 Ptolemaeus und die Karten der arabischen Geographen Mitteil D K K Geogr Ges In Wien 58 152 Mzik Hans von 1916 Afrika nach der arabischen Bearbeitung der gewgrafikὴ ὑfhghsis des Cl Ptolomeaus von Muh ibn Musa al Hwarizmi Denkschriften D Akad D Wissen In Wien Phil hist Kl 59 Mzik Hans von 1926 Das Kitab Ṣurat al Arḍ des Abu Ǧa far Muḥammad ibn Musa al Ḫuwarizmi Leipzig a href Template Cite book html title Template Cite book cite book a CS1 maint location missing publisher link Nallino C A 1896 Al Ḫuwarizmi e il suo rifacimento della Geografia di Tolemo Atti della R Accad Dei Lincei Arno 291 Serie V Memorie Classe di Sc Mor Vol II Rome Ruska Julius 1918 Neue Bausteine zur Geschichte der arabischen Geographie Geographische Zeitschrift 24 77 81 Spitta Wilhelm 1879 Huwarazmi s Auszug aus der Geographie des Ptolemaios Zeitschrift der Deutschen Morgenlandischen Gesellschaft 33 294 297 Archived from the original on 24 June 2021 Retrieved 24 June 2021 General references For a more extensive bibliography see History of mathematics Mathematics in medieval Islam and Astronomy in medieval Islam Berggren John L 2016 Episodes in the Mathhematics of Medieval Islam 2nd ed New York Springer ISBN 978 1 4939 3778 3 Dallal Ahmad 1999 Science Medicine and Technology The Making of a Scientific Culture In Esposito John L ed The Oxford History of Islam New York Oxford University Press ISBN 978 0 19 510799 9 King David A 1999a Islamic Astronomy In Walker Christopher ed Astronomy before the telescope British Museum Press pp 143 174 ISBN 978 0 7141 2733 0 Archived from the original on 25 June 2021 Retrieved 24 June 2021 King David A 2002 A Vetustissimus Arabic Text on the Quadrans Vetus Journal for the History of Astronomy 33 112 237 255 doi 10 1177 002182860203300302 S2CID 125329755 Archived from the original on 25 June 2021 Retrieved 24 June 2021 O Connor John J Robertson Edmund F Abraham bar Hiyya Ha Nasi MacTutor History of Mathematics Archive University of St Andrews O Connor John J Robertson Edmund F Arabic mathematics forgotten brilliance MacTutor History of Mathematics Archive University of St Andrews Rashed Roshdi Armstrong Angela F W 1994 The Development of Arabic Mathematics Between Arithmetic and Algebra New York Springer ISBN 978 90 481 4338 2 Portals nbsp Biography nbsp Mathematics nbsp Geography nbsp Astronomy nbsp Stars nbsp Outer space nbsp Solar System nbsp Science Retrieved from https en wikipedia org w index php title Al Khwarizmi amp oldid 1180114891, wikipedia, wiki, book, books, library,

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