Al-Khwarizmi

9th-century Persian polymath

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

Muḥammad ibn Mūsā al-Khwārizmī

محمد بن موسى الخوارزمي

Woodcut panel depicting al-Khwarizmi, 20th century

Born| c. 780

Khwarazm, Abbasid Caliphate

Died| c. 850[2][3] (aged ~70)

Abbasid Caliphate

Nationality| Persian
Occupation| Head of the House of Wisdom in Baghdad (appt. c. 820)
Academic work
Era| Islamic Golden Age
Main interests|

* Mathematics
* astronomy
* geography

Notable works|

* Al-Jabr (820)
* Zij as-Sindhind (820)
* Kitab Surat al-Ard (833)

Notable ideas| Treatises on algebra and the Hindu–Arabic numeral system
Influenced| Abu Kamil of Egypt[1]

Muhammad ibn Musa al-Khwarizmi[note 1] (Arabic: محمد بن موسى الخوارزمي; c. 780 – c. 850), often referred to as simply al-Khwarizmi, was a Persian polymath who produced vastly influential Arabic-language works in mathematics, astronomy, and geography. Hailing from Khwarazm, he was appointed as the astronomer and head of the House of Wisdom in the city of Baghdad around 820 CE.

His popularizing treatise on algebra, compiled between 813–33 as Al-Jabr (The Compendious Book on Calculation by Completion and Balancing),[6]: 171 presented the first systematic solution of linear and quadratic equations. One of his achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications.[7]: 14 Because al-Khwarizmi was the first person to treat algebra as an independent discipline and introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation),[8] he has been described as the father[9][10][11] or founder[12][13] of algebra. The English term algebra comes from the short-hand title of his aforementioned treatise (الجبر Al-Jabr, transl. "completion" or "rejoining").[14] His name gave rise to the English terms algorism and algorithm; the Spanish, Italian, and Portuguese terms algoritmo; and the Spanish term guarismo[15] and Portuguese term algarismo, both meaning "digit".[16]

In the 12th century, Latin-language translations of al-Khwarizmi's textbook on Indian arithmetic (Algorithmo de Numero Indorum), which codified the various Indian numerals, introduced the decimal-based positional number system to the Western world.[17] Likewise, Al-Jabr, translated into Latin by the English scholar Robert of Chester in 1145, was used until the 16th century as the principal mathematical textbook of European universities.[18][19][20][21]

Al-Khwarizmi revised Geography, the 2nd-century Greek-language treatise by the Roman polymath Claudius Ptolemy, listing the longitudes and latitudes of cities and localities.[22]: 9 He further produced a set of astronomical tables and wrote about calendric works, as well as the astrolabe and the sundial.[23] Al-Khwarizmi made important contributions to trigonometry, producing accurate sine and cosine tables and the first table of tangents.

## Life[edit]

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.[24][25][26] Of Persian stock,[27][24][28][29][30] his name means 'of Khwarazm', a region that was part of Greater Iran,[31] and is now part of Turkmenistan and Uzbekistan.[32]

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),[33] near Baghdad. However, Roshdi Rashed denies this:[34]

> 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.[35]

Regarding al-Khwārizmī's religion, Toomer writes:[36]

> 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 Al-Fihrist includes a short biography on al-Khwārizmī together with a list of his books. Al-Khwārizmī accomplished most of his work between 813 and 833. After the Muslim conquest of Persia, Baghdad had become the centre of scientific studies and trade. Around 820 CE, he was appointed as the astronomer and head of the library of the House of Wisdom.[7]: 14 The House of Wisdom was established by the Abbasid Caliph al-Ma'mūn. Al-Khwārizmī studied sciences and mathematics, including the translation of Greek and Sanskrit scientific manuscripts. He was also a historian who is cited by the likes of al-Tabari and Ibn Abi Tahir.[37]

During the reign of al-Wathiq, he is said to have been involved in the first of two embassies to the Khazars.[38] Douglas Morton Dunlop suggests that Muḥammad ibn Mūsā al-Khwārizmī might have been the same person as Muḥammad ibn Mūsā ibn Shākir, the eldest of the three Banū Mūsā brothers.[39]

## Contributions[edit]

A page from al-Khwārizmī's Algebra

Al-Khwārizmī's contributions to mathematics, geography, astronomy, and cartography established the basis for innovation in algebra and trigonometry. His systematic approach to solving linear and quadratic equations led to algebra, a word derived from the title of his book on the subject, Al-Jabr.[40]

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 in Latin as Algoritmi, led to the term "algorithm".[41][42]

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.[43]

He wrote on mechanical devices like the astrolabe[44] and sundial.[23] 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.[45] 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.[46]

### Algebra[edit]

Main article: Al-Jabr

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.

Al-Jabr (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. It 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 range of problems in trade, surveying and legal inheritance.[47] 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 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.[48]

It provided an exhaustive account of solving polynomial equations up to the second degree,[49] 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.[50]

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.[47]

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}
100 + x 2 − 20 x = 81 x {\displaystyle 100+x^{2}-20x=81x}
x 2 + 100 = 101 x {\displaystyle x^{2}+100=101x}

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}}} ,  p q = 100 {\displaystyle pq=100} 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}}}

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}

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

Solomon 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.[51]

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.[52]

John J. O'Connor and Edmund 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.[53]

Roshdi 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.[54]

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.[55] Regarding the dissimilarity and significance of Al-Khwarizmi's algebraic work from that of Indian Mathematician Brahmagupta, Carl B. 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.[56]

### Arithmetic[edit]

Algorists vs. abacists, depicted in a sketch from 1508 CE Page from a Latin translation, beginning with "Dixit algorizmi"

Al-Khwārizmī's second most influential work was on the subject of arithmetic, which survived in Latin translations but is lost in the original Arabic. His writings include the text kitāb al-ḥisāb al-hindī ('Book of Indian computation'[note 2]), and perhaps a more elementary text, kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī ('Addition and subtraction in Indian arithmetic').[58][59] 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.[60]

As part of 12th century wave of Arabic science flowing into Europe via translations, these texts proved to be revolutionary in Europe.[61] Al-Khwarizmi's Latinized name, Algorismus, turned into the name of method used for computations, and survives in the term "algorithm". It gradually replaced the previous abacus-based methods used in Europe.[62]

Four Latin texts providing adaptions of Al-Khwarizmi's methods have survived, even though none of them is believed to be a literal translation:[58]

* Dixit Algorizmi (published in 1857 under the title Algoritmi de Numero Indorum[63])[64]
* Liber Alchoarismi de Practica Arismetice
* Liber Ysagogarum Alchorismi
* Liber Pulveris

Dixit Algorizmi ('Thus spake Al-Khwarizmi') is the starting phrase of a manuscript in the University of Cambridge library, which is generally referred to by its 1857 title Algoritmi de Numero Indorum. It is attributed to the Adelard of Bath, who had translated the astronomical tables in 1126. It is perhaps the closest to Al-Khwarizmi's own writings.[64]

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.[65]

### Astronomy[edit]

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 as-Sindhind[36] (Arabic: زيج السند هند, "astronomical tables of Siddhanta"[66]) 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.[67] 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.[68]

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 Maslama al-Majriti (c. 1000) has survived in a Latin translation, presumably by Adelard of Bath (26 January 1126).[69] 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[edit]

Al-Khwārizmī's Zīj as-Sindhind contained tables for the trigonometric functions of sines and cosine.[67] A related treatise on spherical trigonometry is attributed to him.[53]

Al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents.[70][71]

### Geography[edit]

Gianluca Gorni's reconstruction of the section of al-Khwārizmī's world map concerning the Indian Ocean. The majority of the placenames used by al-Khwārizmī match those of Ptolemy, 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"),[72] 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.[73]

There is one surviving copy of Kitāb Ṣūrat al-Arḍ, which is kept at the Strasbourg University Library. A Latin translation is at the Biblioteca Nacional de España in Madrid.[74] The book opens with the list of latitudes and longitudes, in order of "weather zones", that is to say in blocks of latitudes and, in each weather zone, by order of longitude. As Paul Gallez notes, this system allows the deduction of many latitudes and longitudes where the only extant document is in such a bad condition, as to make it practically illegible. Neither the Arabic copy nor the Latin translation include the map of the world; however, Hubert Daunicht was able to reconstruct the missing map from the list of coordinates. Daunicht read the latitudes and longitudes of the coastal points in the manuscript, or deduced them from the context where they were not legible. He transferred the points onto graph paper and connected them with straight lines, obtaining an approximation of the coastline as it was on the original map. He did the same for the rivers and towns.[75]

Al-Khwārizmī corrected Ptolemy's gross overestimate for the length of the Mediterranean Sea[76] from the Canary Islands to the eastern shores of the Mediterranean; Ptolemy overestimated it at 63 degrees of longitude, while al-Khwārizmī almost correctly estimated it at nearly 50 degrees of longitude. He "depicted the Atlantic and Indian Oceans as open bodies of water, not land-locked seas as Ptolemy had done."[77] 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.[76]

### Jewish calendar[edit]

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 Al-Bīrūnī and Maimonides.[36]

### Other works[edit]

Ibn al-Nadim's 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.[78]

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 wrote two books on using and constructing astrolabes.

## Honours[edit]

A Soviet postage stamp issued 6 September 1983, commemorating al-Khwārizmī's (approximate) 1200th birthday

* Al-Khwarizmi (crater) — A crater on the far side of the Moon. [79]
* 13498 Al Chwarizmi — Main-belt Asteroid, Discovered 1986 Aug 6 by E. W. Elst and V. G. Ivanova at Smolyan.[80]
* 11156 Al-Khwarismi — Main-belt Asteroid, Discovered 1997 Dec 31 by P. G. Comba at Prescott.[81]

## Notes[edit]

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.[57]

## References[edit]

1. ^ O'Connor, John J.; Robertson, Edmund F., "Abū Kāmil Shujā' ibn Aslam" Archived 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 Archived 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. ^ Oaks, J. (2009), "Polynomials and Equations in Arabic Algebra", Archive for History of Exact Sciences, 63(2), 169–203.
7. ^ a b Maher, P. (1998), "From Al-Jabr to Algebra", Mathematics in School, 27(4), 14–15.
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. Archived 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. ^ Gandz, Solomon, The sources of al-Khwarizmi's algebra, Osiris, i (1936), 263–277, "Al-Khwarizmi's algebra is regarded as the foundation and cornerstone of the sciences. In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers."
12. ^ 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."
13. ^ 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."
14. ^ Brentjes, Sonja (1 June 2007). "Algebra". Encyclopaedia of Islam, THREE. Archived from the original on 22 December 2019. Retrieved 5 June 2019.
15. ^ 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.
16. ^ Gandz, Solomon (1926). "The Origin of the Term "Algebra"". The American Mathematical Monthly. 33 (9): 437–440. doi:10.2307/2299605. ISSN 0002-9890. JSTOR 2299605.
17. ^ Struik 1987, p. 93
18. ^ 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.
19. ^ 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"
20. ^ Shawn Overbay; Jimmy Schorer; Heather Conger. "Al-Khwarizmi". University of Kentucky. Archived from the original on 12 December 2013.
21. ^ "Islam Spain and the history of technology". www.sjsu.edu. Archived from the original on 11 October 2018. Retrieved 24 January 2018.
22. ^ van der Waerden, Bartel Leendert (1985). A History of Algebra: From al–Khwarizmi to Emmy Noether. Berlin: Springer-Verlag.
23. ^ a b Arndt 1983, p. 669
24. ^ 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."
25. ^ 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. Archived 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.
26. ^ Dodge, Bayard, ed. (1970), The Fihrist of al-Nadīm: A Tenth-Century Survey of Islamic Culture, vol. 2, translated by Dodge, New York: Columbia University Press
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28. ^ 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)."
29. ^ 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"
30. ^ 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."
31. ^ Encycloaedia Iranica-online, s.v. "CHORASMIA, ii. In Islamic times Archived 2 September 2021 at the Wayback Machine," by Clifford E. Bosworth.
32. ^ Bosworth, Clifford Edmund (1960–2005). "Khwārazm". 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. 1060–1065. OCLC 399624.
33. ^ "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
34. ^ Rashed, Roshdi (1988). "al-Khwārizmī's Concept of Algebra". In Zurayq, Qusṭanṭīn; 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.
35. ^ King, David A. (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. 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."
36. ^ a b c Toomer 1990
37. ^ 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.
38. ^ 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.
39. ^ Dunlop 1943
40. ^ 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.
41. ^ Daffa 1977
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48. ^ 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. Archived from the original on 30 October 2020. Retrieved 29 September 2020.
49. ^ 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."
50. ^ (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."
51. ^ Gandz, Solomon, The sources of al-Khwarizmi's algebra, Osiris, i (1936), 263–277
52. ^ 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.
53. ^ 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
54. ^ Rashed, R.; Armstrong, Angela (1994). The Development of Arabic Mathematics. Springer. pp. 11–12. ISBN 978-0-7923-2565-9. OCLC 29181926.
55. ^ 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."
56. ^ Boyer, Carl Benjamin (1968). A History of Mathematics. p. 252.
57. ^ 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, 57 (4), The University of Chicago Press: 475–490, doi:10.1086/350163, JSTOR 228518, S2CID 143979243
58. ^ a b Burnett 2017, p. 39.
59. ^ 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
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63. ^ "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
64. ^ a b Crossley, John N.; Henry, Alan S. (1990), "Thus Spake al-Khwārizmī: 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
65. ^ "How Algorithm Got Its Name". earthobservatory.nasa.gov. 8 January 2018.
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67. ^ a b Kennedy 1956, pp. 26–29
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69. ^ Kennedy 1956, p. 128
70. ^ 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.
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72. ^ 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".
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### Sources[edit]

* Arndt, 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 & 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 & Francis, ISBN 978-1-351-67617-5, archived from the original on 28 March 2023, retrieved 5 May 2019
* Daffa, Ali Abdullah al- (1977). The Muslim contribution to mathematics. London: Croom Helm. ISBN 978-0-85664-464-1.
* 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. 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, Régis (1996), Encyclopedia of the History of Arabic Science, vol. 1, Routledge, ISBN 0-415-12410-7
* Struik, Dirk Jan (1987). A Concise History of Mathematics (4th ed.). Dover Publications. ISBN 978-0-486-60255-4.
* Toomer, Gerald (1990). "Al-Khwārizmī, Abu Ja'far Muḥammad ibn Mūsā". 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 reading[edit]

### Biographical[edit]

* Brentjes, Sonja (2007). "Khwārizmī: Muḥammad ibn Mūsā al‐Khwārizmī 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, F., ed., Islamic Mathematics and Astronomy, Frankfurt: Institut für Geschichte der arabisch-islamischen Wissenschaften, 1997–99.

### Algebra[edit]

* 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-Khowārizmī'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-Khuwārizmī". 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-Khwārizmī's al-Jabr, A Critical Edition". Mediaeval Studies. 48: 211–263. doi:10.1484/J.MS.2.306339.
* Hughes, Barnabas. Robert of Chester's Latin translation of al-Khwarizmi's al-Jabr: A new critical edition. In Latin. F. Steiner Verlag Wiesbaden (1989). ISBN 3-515-04589-9.
* 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.

### Astronomy[edit]

* 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. 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-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. 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-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.
* 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. Archived from the original on 24 June 2021. Retrieved 24 June 2021. (Van Dalen's homepage. List of Publications, Articles – no. 5).

### Jewish calendar[edit]

* Kennedy, E. S. (1964). "Al-Khwārizmī on the Jewish Calendar". Scripta Mathematica. 27: 55–59.

## External links[edit]

Wikiquote has quotations related to al-Khwārizmī.

* Media related to Muhammad ibn Musa al-Khwarizmi at Wikimedia Commons

Articles and topics related to al-Khwarizmi

* v
* t
* e

Mathematics in the medieval Islamic world

Mathematicians|

| 9th century|

* 'Abd al-Hamīd ibn Turk
* Sanad ibn Ali
* al-Jawharī
* Al-Ḥajjāj ibn Yūsuf
* Al-Kindi
* Qusta ibn Luqa
* Al-Mahani
* al-Dinawari
* Banū Mūsā brothers
* Hunayn ibn Ishaq
* Al-Khwarizmi
* Yusuf al-Khuri
* Ishaq ibn Hunayn
* Na'im ibn Musa
* Thābit ibn Qurra
* al-Marwazi
* Abu Said Gorgani

|

10th century|

* Abu al-Wafa
* al-Khazin
* Al-Qabisi
* Abu Kamil
* Ahmad ibn Yusuf
* Aṣ-Ṣaidanānī
* Sinān ibn al-Fatḥ
* al-Khojandi
* Al-Nayrizi
* Al-Saghani
* Brethren of Purity
* Ibn Sahl
* Ibn Yunus
* al-Uqlidisi
* Al-Battani
* Sinan ibn Thabit
* Ibrahim ibn Sinan
* Al-Isfahani
* Nazif ibn Yumn
* al-Qūhī
* Abu al-Jud
* Al-Sijzi
* Al-Karaji
* al-Majriti
* al-Jabali

11th century|

* Abu Nasr Mansur
* Alhazen
* Kushyar Gilani
* Al-Biruni
* Ibn al-Samh
* Abu Mansur al-Baghdadi
* Avicenna
* al-Jayyānī
* al-Nasawī
* al-Zarqālī
* ibn Hud
* Al-Isfizari
* Omar Khayyam
* Muhammad al-Baghdadi

12th century|

* Jabir ibn Aflah
* Al-Kharaqī
* Al-Khazini
* Al-Samawal al-Maghribi
* al-Hassar
* Sharaf al-Din al-Tusi
* Ibn al-Yasamin

13th century|

* Ibn al‐Ha'im al‐Ishbili
* Ahmad al-Buni
* Ibn Munim
* Alam al-Din al-Hanafi
* Ibn Adlan
* al-Urdi
* Nasir al-Din al-Tusi
* al-Abhari
* Muhyi al-Din al-Maghribi
* al-Hasan al-Marrakushi
* Qutb al-Din al-Shirazi
* Shams al-Din al-Samarqandi
* Ibn al-Banna'
* Kamāl al-Dīn al-Fārisī

14th century|

* Nizam al-Din al-Nisapuri
* Ibn al-Shatir
* Ibn al-Durayhim
* Al-Khalili
* al-Umawi

15th century|

* Ibn al-Majdi
* al-Rūmī
* al-Kāshī
* Ulugh Beg
* Ali Qushji
* al-Wafa'i
* al-Qalaṣādī
* Sibt al-Maridini
* Ibn Ghazi al-Miknasi

16th century|

* Al-Birjandi
* Muhammad Baqir Yazdi
* Taqi ad-Din
* Ibn Hamza al-Maghribi

Mathematical
works|

* The Compendious Book on Calculation by Completion and Balancing
* De Gradibus
* Principles of Hindu Reckoning
* Book of Optics
* The Book of Healing
* Book on the Measurement of Plane and Spherical Figures
* Encyclopedia of the Brethren of Purity
* Toledan Tables
* Tabula Rogeriana
* Zij

Concepts|

* Alhazen's problem
* Islamic geometric patterns

Centers|

* Al-Azhar University
* Al-Mustansiriya University
* House of Knowledge
* House of Wisdom
* Constantinople observatory of Taqi ad-Din
* Madrasa
* Maragheh observatory
* University of al-Qarawiyyin

Influences|

* Babylonian mathematics
* Greek mathematics
* Indian mathematics

Influenced|

* Byzantine mathematics
* European mathematics
* Indian mathematics

Related|

* Hindu–Arabic numeral system
* Arabic numerals (Eastern Arabic numerals, Western Arabic numerals)
* Trigonometric functions
* History of trigonometry
* History of algebra

* v
* t
* e

Mathematics in Iran

Mathematicians|

| Before
20th Century|

* Abu al-Wafa' Buzjani
* Jamshid al-Kashi (al-Kashi's theorem)
* Omar Khayyam (Khayyam-Pascal's triangle, Khayyam-Saccheri quadrilateral, Khayyam's Solution of Cubic Equations)
* Al-Mahani
* Muhammad Baqir Yazdi
* Nizam al-Din al-Nisapuri
* Al-Nayrizi
* Kushyar Gilani
* Ayn al-Quzat Hamadani
* Al-Isfahani
* Al-Isfizari
* Al-Khwarizmi (Al-jabr)
* Najm al-Din al-Qazwini al-Katibi
* Nasir al-Din al-Tusi
* Al-Biruni

|

Modern|

* Maryam Mirzakhani
* Caucher Birkar
* Sara Zahedi
* Farideh Firoozbakht (Firoozbakht's conjecture)
* S. L. Hakimi (Havel–Hakimi algorithm)
* Siamak Yassemi
* Freydoon Shahidi (Langlands–Shahidi method)
* Hamid Naderi Yeganeh
* Esmail Babolian
* Ramin Takloo-Bighash
* Lotfi A. Zadeh (Fuzzy mathematics, Fuzzy set, Fuzzy logic)
* Ebadollah S. Mahmoodian
* Reza Sarhangi (The Bridges Organization)
* Siavash Shahshahani
* Gholamhossein Mosaheb
* Amin Shokrollahi
* Reza Sadeghi
* Mohammad Mehdi Zahedi
* Mohsen Hashtroodi
* Hossein Zakeri
* Amir Ali Ahmadi

Prize Recipients|

| Fields Medal|

* Maryam Mirzakhani (2014)
* Caucher Birkar (2018)

|

EMS Prize|

* Sara Zahedi (2016)

Satter Prize|

* Maryam Mirzakhani (2013)

Organizations|

* Iranian Mathematical Society

Institutions|

* Institute for Research in Fundamental Sciences

* v
* t
* e

Astronomy in the medieval Islamic world

Astronomers

* by century

8th|

* Ahmad Nahavandi
* Al-Fadl ibn Naubakht
* Muḥammad ibn Ibrāhīm al-Fazārī
* Ibrāhīm al-Fazārī
* Mashallah ibn Athari
* Yaʿqūb ibn Ṭāriq

9th|

* Abu Ali al-Khayyat
* Abu Ma'shar al-Balkhi
* Abu Said Gorgani
* Al-Farghani
* Al-Kindi
* Al-Mahani
* Abu Hanifa Dinawari
* Al-Ḥajjāj ibn Yūsuf
* Al-Marwazi
* Ali ibn Isa al-Asturlabi
* Banū Mūsā brothers
* Iranshahri
* Khalid ibn Abd al‐Malik al‐Marwarrudhi
* Al-Khwarizmi
* Sahl ibn Bishr
* Thābit ibn Qurra
* Yahya ibn Abi Mansur

10th|

* al-Sufi
* Ibn
* Al-Adami
* al-Khojandi
* al-Khazin
* al-Qūhī
* Abu al-Wafa
* Ahmad ibn Yusuf
* al-Battani
* Al-Qabisi
* Ibn al-A'lam
* Al-Nayrizi
* Al-Saghani
* Aṣ-Ṣaidanānī
* Ibn Yunus
* Ibrahim ibn Sinan
* Ma Yize
* al-Sijzi
* Al-ʻIjliyyah
* Nastulus
* Abolfadl Harawi
* Haseb-i Tabari
* al-Majriti
* Abu al-Hasan al-Ahwazi

11th|

* Abu Nasr Mansur
* al-Biruni
* Ali ibn Ridwan
* Al-Zarqālī
* Ibn al-Samh
* Alhazen
* Avicenna
* Ibn al-Saffar
* Kushyar Gilani
* Said al-Andalusi
* Ibrahim ibn Said al-Sahli
* Ibn Mu'adh al-Jayyani
* Al-Isfizari
* Ali ibn Khalaf

12th|

* Al-Bitruji
* Avempace
* Ibn Tufail
* Al-Kharaqī
* Al-Khazini
* Al-Samawal al-Maghribi
* Abu al-Salt
* Averroes
* Ibn al-Kammad
* Jabir ibn Aflah
* Omar Khayyam
* Sharaf al-Din al-Tusi

13th|

* Ibn al-Banna' al-Marrakushi
* Ibn al‐Ha'im al‐Ishbili
* Jamal ad-Din
* Alam al-Din al-Hanafi
* Najm al‐Din al‐Misri
* Muhyi al-Din al-Maghribi
* Nasir al-Din al-Tusi
* Qutb al-Din al-Shirazi
* Shams al-Din al-Samarqandi
* Zakariya al-Qazwini
* al-Urdi
* al-Abhari
* Muhammad ibn Abi Bakr al‐Farisi
* Abu Ali al-Hasan al-Marrakushi
* Ibn Ishaq al-Tunisi
* Ibn al‐Raqqam
* Al-Ashraf Umar II
* Fakhr al-Din al-Akhlati

14th|

* Ibn al-Shatir
* Al-Khalili
* Ibn Shuayb
* al-Battiwi
* Abū al‐ʿUqūl
* Al-Wabkanawi
* Nizam al-Din al-Nisapuri
* al-Jadiri
* Sadr al-Shari'a al-Asghar
* Fathullah Shirazi

15th|

* Ali Kuşçu
* Abd al‐Wajid
* Jamshid al-Kashi
* Kadızade Rumi
* Ulugh Beg
* Sibt al-Maridini
* Ibn al-Majdi
* al-Wafa'i
* al-Kubunani
* 'Abd al-'Aziz al-Wafa'i

16th|

* Al-Birjandi
* al-Khafri
* Baha' al-din al-'Amili
* Piri Reis
* Takiyüddin

17th|

* Yang Guangxian
* Ehmedê Xanî
* Al Achsasi al Mouakket
* Muhammad al-Rudani

Topics

| Works|

* Arabic star names
* Islamic calendar
* Aja'ib al-Makhluqat
* Encyclopedia of the Brethren of Purity
* Tabula Rogeriana
* The Book of Healing
* The Remaining Signs of Past Centuries

Zij|

* Alfonsine tables
* Huihui Lifa
* Book of Fixed Stars
* Toledan Tables
* Zij-i Ilkhani
* Zij-i Sultani
* Sullam al-sama'

Instruments|

* Alidade
* Analog computer
* Aperture
* Armillary sphere
* Astrolabe
* Astronomical clock
* Celestial globe
* Compass
* Compass rose
* Dioptra
* Equatorial ring
* Equatorium
* Globe
* Graph paper
* Mural instrument
* Navigational astrolabe
* Octant
* Planisphere
* Quadrant
* Sextant
* Shadow square
* Sundial
* Schema for horizontal sundials
* Triquetrum

Concepts|

* Almucantar
* Apogee
* Astrology
* Astrophysics
* Axial tilt
* Azimuth
* Celestial mechanics
* Celestial spheres
* Circular orbit
* Deferent and epicycle
* Earth's rotation
* Eccentricity
* Ecliptic
* Elliptic orbit
* Equant
* Galaxy
* Geocentrism
* Gravitational energy
* Gravity
* Heliocentrism
* Inertia
* Islamic cosmology
* Moonlight
* Multiverse
* Muwaqqit
* Obliquity
* Parallax
* Precession
* Qibla
* Salah times
* Specific gravity
* Spherical Earth
* Sublunary sphere
* Sunlight
* Supernova
* Temporal finitism
* Trepidation
* Triangulation
* Tusi couple
* Universe

Institutions|

* Al-Azhar University
* House of Knowledge
* House of Wisdom
* University of al-Qarawiyyin
* Observatories
* Constantinople (Taqi al-Din)
* Maragheh
* Samarkand (Ulugh Beg)

Influences|

* Babylonian astronomy
* Egyptian astronomy
* Hellenistic astronomy
* Indian astronomy

Influenced|

* Byzantine science
* Chinese astronomy
* Medieval European science
* Indian astronomy

* v
* t
* e

Geography and cartography in the medieval Islamic world

Geographers|

| 9th century|

* Al-Khwarizmi
* Abu Hanifa Dinawari
* Ya'qubi
* Sulaiman al-Tajir

|

10th century|

* Ibn Khordadbeh
* Ahmad ibn Rustah
* Ahmad ibn Fadlan
* Abu Zayd al-Balkhi
* Abu Muhammad al-Hasan al-Hamdani
* Al-Masudi
* Istakhri
* Khashkhash Ibn Saeed Ibn Aswad
* Ibn Hawqal
* Ibn al-Faqih
* Al-Muqaddasi
* Al-Ramhormuzi
* Qudama ibn Ja'far

11th century|

* Al-Biruni
* Abu Saʿīd Gardēzī
* Al-Bakri
* Mahmud al-Kashgari
* Domiyat

12th century|

* al-Zuhri
* Muhammad al-Idrisi
* Abu'l Abbas al-Hijazi

13th century|

* Ibn Jubayr
* Saadi Shirazi
* Yaqut al-Hamawi
* Ibn Said al-Maghribi
* Ibn al-Nafis
* Ibn al-Mujawir

14th century|

* Al-Dimashqi
* Abu'l-Fida
* Ibn al-Wardi
* Hamdallah Mustawfi
* Ibn Battuta
* Lin Nu

15th century|

* Abd-al-Razzāq Samarqandī
* Ghiyāth al-dīn Naqqāsh
* Ahmad ibn Mājid
* Zheng He
* Ma Huan
* Fei Xin

16th century|

* Sulaiman Al Mahri
* Piri Reis
* Mir Ahmed Nasrallah Thattvi
* Amīn Rāzī

17th century|

* Evliya Çelebi

Works|

* Book of Roads and Kingdoms (al-Bakrī)
* Book of Roads and Kingdoms (ibn Khordadbeh)
* Tabula Rogeriana
* Kitab al-Rawd al-Mitar
* Mu'jam Al-Buldan
* Rihla
* The Meadows of Gold
* Piri Reis map
* Kitab al-Kharaj

Influences|

* Geography (Ptolemy)

* v
* t
* e

People of Khorasan

Scientists|

* Abu Hatam Isfizari
* Abu Ma'shar al-Balkhi
* Abu Wafa
* Abu Ubayd Juzjani
* Abu Zayd Balkhi
* Alfraganus
* Ali Qushji
* Avicenna
* Birjandi
* Biruni
* Hasib Marwazi
* Ibn Hayyan
* Abu Ja'far al-Khazin
* Khazini
* Khojandi
* Khwarizmi
* Nasawi
* Nasir al-Din al-Tusi
* Omar Khayyam
* Sharaf al-Din al-Tusi
* Sijzi

Philosophers|

* Algazel
* Amiri
* Avicenna
* Farabi
* Haji Bektash Veli
* Nasir Khusraw
* Sijistani
* Shahrastani

Islamic scholars|

* Abu Dawud al-Sijistani
* Abu Barakat Nasafi
* Abu Hanifa
* Abu Hafs Nasafi
* Abu Layth Samarqandi
* Abu Mu'in Nasafi
* Abu Qasim Samarqandi
* Ansari
* Baghavi
* Bayhaqi
* Bazdawi
* Bukhari
* Dabusi
* Fatima Samarqandi
* Ghazali
* Ghaznawi
* Hakim Tirmidhi
* Hakim Nishapuri
* Ibn Hibban
* Ibn Mubarak
* Ibn Tayfour Sajawandi
* Juwayni
* Kasani
* Kashifi
* Lamishi
* Marghinani
* Maturidi
* Mulla al-Qari
* Muqatil
* Muslim
* Nasa'i
* Qushayri
* Razi
* Sabuni
* Sajawandi
* Sarakhsi
* Shaykh Tusi
* Taftazani
* Tha'labi Nishapuri
* Tirmidhi
* Zamakhshari

Poets and artists|

* Abu Sa'id Abu'l-Khayr
* Anvari
* Aruzi Samarqandi
* Asadi Tusi
* Attar Nishapuri
* Behzad
* Daqiqi
* Farrukhi Sistani
* Ferdowsi
* Jami
* Kashifi
* Nasir Khusraw
* Rabia Balkhi
* Rudaki
* Rumi
* Sanā'ī

Historians and
political figures|

* Abu'l-Fadl Bayhaqi
* Abu'l-Hasan Isfarayini
* Abu'l-Ma'ali Nasrallah
* Abu Muslim Khorasani
* Gardizi
* Ali-Shir Nava'i
* Ata-Malik Juvayni
* Aufi
* Abu Ali Bal'ami
* Gawhar Shad
* Ibn Khordadbeh
* Khalid ibn Barmak
* Minhaj al-Siraj Juzjani
* Nizam al-Mulk
* Tahir ibn Husayn
* Yahya Barmaki
* Ahmad ibn Nizam al-Mulk
* Shihab al-Nasawi

Portals:

* Biography
* Mathematics
* Geography
* Astronomy
* Stars
* Outer space
* Solar System
* Science

Authority control databases

International|

* FAST
* ISNI
* 2
* 3
* VIAF
* 2
* 3
* 4
* WorldCat
* 2

National|

* Norway
* 2
* Spain
* 2
* France
* BnF data
* Catalonia
* Germany
* Israel
* United States
* Sweden
* Czech Republic
* Australia
* Netherlands
* 2
* Poland
* Vatican

Academics|

* CiNii
* MathSciNet
* zbMATH

People|

* Deutsche Biographie
* Trove

Other|

* IdRef
* İslâm Ansiklopedisi

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