Muhammad ibn Musa al-Khwarizmi


Abu Abdallah Muḥammad ibn Mūsā al-Jwārizmī (Persian: ابوعبدالله محمد بن موسی جوارزمی (Khorasmia ,ca. 780-Baghdad, ca. 850), generally known as al-Khwarismi, and formerly Latinized as Algorithmi, was a Persian mathematician, astronomer, and geographer. He was an astronomer and head of the Library of the House of Wisdom in Baghdad, ca. 820. He is considered one of the greatest mathematicians in history.

His work, Compendium of calculus by reintegration and comparison, presented the first systematic solution of linear and quadratic equations. One of his main achievements in the field of algebra was his demonstration of how to solve quadratic equations with the method of completion of squares, justifying it geometrically. He also worked in the field of trigonometry, producing tables of sine and cosine, and the first on tangents.

His importance lies in the fact that he was the first to treat algebra as an independent discipline and introduced the methods of “reduction” and “equilibrium”, being described as the father and founder of algebra. In fact, his Latinized name gave name to several mathematical terms such as algoritmo and algoritmia (the discipline that develops algorithms) and the Portuguese algarismo which means digit, as well as guarismo.

He also excelled as a geographer and astronomer, revising Ptolemy”s work, Geography, and managing to enumerate longitudes and latitudes of several cities and localities. He also wrote several works on the astrolabe, the sundial, the calendar, and produced several astronomical tables.

His legacy continued when in the 12th century Latin translations of his work Algoritmi de número Indorum helped popularize Arabic numerals in the West, along with the work of the Italian mathematician Fibonacci, leading to the replacement of the Roman numbering system with Arabic, which gave rise to the current numeration. In addition, his magnum opus was used as the main treatise on mathematics, translated by Robert of Chester in 1145, in European universities until the 16th century.

Little is known about his biography, to such an extent that there are unresolved discussions about his place of birth. Some maintain that he was born in Baghdad. Others, following Gerald Toomer”s article (based on the writings of the historian al-Tabari) maintain that he was born in the Khorasmian city of Khiva (in present-day Uzbekistan). Rashed finds that this is an error of interpretation by Toomer, due to a transcription error (the lack of the connective wa) in a copy of al-Tabari”s manuscript. This will not be the last disagreement between historians that we will find in the descriptions of al-Khwarismi”s life and works. He studied and worked in Baghdad in the first half of the ninth century, at the court of the caliph al-Mamun. For many, he was the greatest mathematician of his time.

We owe to his name and that of his main work, Hisāb al-ŷabr wa”l muqābala, (حساب الجبر و المقابلة) our words algebra, guarism and algorithm. In fact, he is considered as the father of algebra and as the introducer of our numbering system called Arabic.

Around 815 al-Mamun, the seventh Abbasid caliph, son of Harun al-Rashid, founded in his capital, Baghdad, the House of Wisdom (Bayt al-Hikma), a research and translation institution that some have compared to the Library of Alexandria. Greek and Hindu scientific and philosophical works were translated into Arabic. It also had astronomical observatories. It was in this scientific and multicultural environment that al-Khwarismi was educated and worked along with other scientists such as the Banu Musa brothers, al-Kindi and the famous translator Hunayn ibn Ishaq. Two of his works, his treatises on algebra and astronomy, are dedicated to the caliph himself.


In his algebra treatise Hisāb al-ŷabr wa”l muqābala (حساب الجبر و المقابلة, Compendium of Calculus by Completion and Comparison), an eminently didactic work, is intended to teach an algebra applied to solving problems of everyday life in the Islamic empire of the time. Rosen”s translation of al-Khwarismi”s words describing the aims of his book show that the scholar intended to teach:

… that which is easy and most useful in arithmetic, such as men constantly require in cases of inheritance, bequests, partitions, trials, and commerce, and in all their dealings with each other, or when it comes to the measurement of land, the digging of canals, geometrical calculations, and other objects of various kinds and types.

Translated into Latin by Gerardo de Cremona in Toledo, it was used in European universities as a textbook until the sixteenth century, being the first known treatise in which an exhaustive study is made on the resolution of equations.

After introducing the natural numbers, al-Khwarismi addresses the main issue in the first part of the book: the solution of equations. His equations are linear or quadratic and are composed of units, roots, and squares; for him, for example, a unit was a number, a root was x {displaystyle x} and a square x 2 {displaystyle x^{2}} . Although in the examples that follow we will use the algebraic notation common in our day to help the reader understand the notions, it should be noted that al-Khwarizmi did not use symbols of any kind, but only words.

First reduce an equation to one of six normal forms:

The reduction is carried out using the operations of al-ŷabr (“completion”, the process of eliminating negative terms from the equation) and al-muqabala (“balancing”, the process of reducing positive terms of the same power when they occur on both sides of the equation). Then, al-Khwarismi shows how to solve the six types of equations, using algebraic and geometric methods of solution. For example, to solve the equation x 2 + 10 x = 39 { “displaystyle x^{2}+10x=39} , write:

… a square and ten roots equal 39 units. So, the question in this type of equation is roughly like this: what is the square that, combined with ten of its roots, will give a sum total of 39. The way to solve this type of equation is to take half of the roots mentioned. Now, the roots in the problem before us are ten. Therefore, we take 5 which multiplied by itself gives 25, a quantity which you will add to 39 giving 64. Having extracted the square root of this, which is 8, we subtract therefrom half of the roots, 5, resulting in 3.

There follows the geometrical proof by completion of the square, which we will not discuss here. We will point out however that the geometrical proofs used by al-Khwarismi are the subject of controversy among scholars. The question, which remains unanswered, is whether he was familiar with Euclid”s work. It should be remembered, in al-Khwarismi”s youth and during the reign of Harun al-Rashid, al-Hajjaj had translated the Elements into Arabic, and was one of al-Khwarismi”s companions in the House of Wisdom. This would support Toomer”s position (op.cit.). Rashed comments that he was probably inspired by the recent knowledge of “the Elements”. But, for his part, Gandz maintains that the Elements were completely unknown to him. Although it is uncertain whether he actually knew the Euclidean work, it is possible to claim that he was influenced by other works of geometry; see Parshall”s treatment of the methodological similarities with the Hebrew text Mishnat ha Middot, of the mid-second century.

He continues Hisab al-ŷabr wa”l-muqabala by examining how the laws of arithmetic extend to its algebraic objects. For example, he shows how to multiply expressions such as. ( a + b x ) ( c + d x ) { displaystyle (a+bx)(c+dx)} . Rashed (op. cit.) finds his forms of resolution extremely original, but Crossley considers them less significant. Gandz considers that the paternity of algebra is much more attributable to al-Khwarismi than to Diophantus.

The next part consists of applications and examples. It describes rules for finding the area of geometric figures such as the circle, and the volume of solids such as the sphere, cone and pyramid. This section certainly has much greater affinity with Hebrew and Indian texts than with any Greek work. The final part of the book deals with the complex Islamic rules of inheritance, but requires little of the algebra he expounded earlier, beyond solving linear equations.


Of his arithmetic, possibly originally called Kitab al-Ŷamaa wa al-Tafriq bi Hisab al-Hind, (كتاب الجامع و التفريق بحساب الهند), Book of addition and subtraction, according to the Indian calculus, we only preserve a Latin version from the 12th century, Algoritmi de numero Indorum and another entitled Liber Algoarismi translated by Juan Hispalense, belonging to the Toledan School of Translation, in 1133. Unfortunately, it is known that the work deviates quite a bit from the original text. This work describes in detail the Indo-Arabic numerals, the Indian positional numbering system in base 10 and methods for making calculations with it. It is known that there was a method for finding square roots in the Arabic version, but it does not appear in the Latin version. It was possibly the first to use zero as a positional indicator. He was essential for the introduction of this numbering system in the Arab world, al-Andalus and later in Europe. André Allard discusses some 12th century Latin treatises based on this lost work.

As part of the 12th century wave of Arabic science that flowed into Europe through translations, these texts proved to be revolutionary in Europe. Al-Khwarizmi”s Latinized name, Algorismus, became the name of the method used for the calculations and survives in the modern term “algorithm”. It gradually replaced the earlier abacus-based methods used in Europe.

Four Latin texts have survived that provide adaptations of Al-Khwarizmi”s methods, although none of them is believed to be a literal translation.

Dixit Algorizmi (”Thus spoke Al-Khwarizmi”) is the opening sentence of a manuscript in the Cambridge University Library, generally referred to by its 1857 title Algoritmi de Numero Indorum. It is attributed to Adelard of Bath, who had also translated the astronomical tables in 1126. It is perhaps closest to Al-Khwarizmi”s own writings.

Al-Khwarizmi”s work on arithmetic was responsible for introducing Arabic numerals, based on the Hindu-Arabic numbering system developed in Indian mathematics, to the Western world. The term “algorithm” is derived from algorithm, the technique of performing arithmetic with Indo-Arabic numerals developed by al-Khwarizmi. Both “algorithm” and “algorism” are derived from the Latinized forms of al-Khwwārizmī”s name, Algoritmi and Algorismi , respectively.


Of his treatise on astronomy, Sindhind zij, the two versions he wrote in Arabic have also been lost. This work is based on Indian astronomical works “unlike later Islamic manuals of astronomy, which used the Greek planetary models of Ptolemy”s ”Almagest.”” The Indian text on which the treatise is based is one of those gifted to the Baghdad court around 770 by a diplomatic mission from India. In the 10th century al-Maŷriti made a critical revision of the shorter version, which was translated into Latin by Adelard of Bath; there is also a Latin translation of the longer version, and both translations have come down to our own time. The main topics covered in the work are calendars; the calculation of the true positions of the Sun, Moon, and planets; tables of sines and tangents; spherical astronomy; astrological tables; calculations of parallaxes and eclipses; and visibility of the Moon. Rozenfel”d discusses a related manuscript on spherical trigonometry, attributed to al-Khwarismi.


In the field of geography, in a work called Kitab Surat al-Ard (Arabic: كتاب صورةلأرض ,Book of the Appearance of the Earth or the Image of the Earth), written in 833, he revised and corrected Ptolemy”s earlier works regarding Africa and the Orient. It lists latitudes and longitudes of 2,402 places, and placed cities, mountains, seas, islands, geographical regions and rivers as the basis for a map of the then known world. It includes maps that, on the whole, are more accurate than those of Ptolemy. It is clear that where there was greater local knowledge available to al-Khwârazm, such as the regions of Islam, Africa and the Far East, the work is much more accurate than Ptolemy”s, but he seems to have used Ptolemy”s data for Europe. Seventy geographers are said to have worked under him on these maps.

There is only a single surviving copy of the Kitab Surat-al-Ard, kept in the Library of the University of Strasbourg. A copy translated into Latin is kept at the National Library of Spain in Madrid.

Although neither the Arabic copy nor the Latin translation includes the world map, Hubert Daunicht was able to reconstruct a world map using his list of coordinates.

Al-Khwarizmi corrected Ptolemy”s overestimate of the surface area of the Mediterranean Sea (Ptolemy made an estimate that the Mediterranean Sea was 63 degrees long, while he made the more correct estimate that the sea was about 50 degrees long. He also contradicted Ptolemy by saying that the Atlantic Ocean and the Indian Ocean were two open bodies of water, not seas. Al-Khwarizmi also set the Old World Greenwich meridian on the eastern shore of the Mediterranean, 10-13 degrees east of Alexandria (Ptolemy placed the meridian 70 degrees west of Baghdad). Most Muslim geographers of the medieval age continued to use al-Khwarizmi”s Greenwich meridian.

Most of the place names used by al-Khwarizmi coincide with those of Ptolemy, Martellus and Behaim. The general shape of the coast is the same between Taprobane and Kattigara. The Atlantic coast of the Tail of the Dragon, which does not exist in Ptolemy”s map, is traced in very little detail in al-Khwarizmi”s map, but it is clear and more precise than that of Martellus” map and Behaim”s version.

Other works

Ibn al-Nadim”s Kitāb al-Fihrist, an index of Arabic books, mentions al-Khwārizmī”s Kitāb al-Taʾrīkh (however, a copy had reached Nusaybin in the 11th century, where it was found by its metropolitan bishop, Mar Elyas bar Shinaya. Elijah”s chronicle quotes him from “the death of the Prophet” to 169 AH, at which time Elijah”s text is found in a lacuna.

Several Arabic manuscripts in Berlin, Istanbul, Tashkent, Cairo, and Paris contain further material that surely or with some probability comes from al-Khwwārizmī. The Istanbul manuscript contains an article on sundials; the fihrist attributes al-Khwwārizmī Kitāb ar-Rukhāma (t) ( Arabic : كتاب الرخامة ). Other works, such as one on determining the direction of Mecca, deal with spherical astronomy

Two texts deserve special interest concerning the width of the morning ( 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 ” ).

His known work is completed with a series of minor works on subjects such as the astrolabe, on which he wrote two texts, on sundials and on the Jewish calendar. He also wrote a political history containing horoscopes of prominent personages.

In Khiva, Uzbekistan, the place often accepted as his probable birthplace, there is a statue in his honor. The image shows Juarismi seated on a bench, in a position of reasoning, as the image looks towards the ground, as if he were calculating or reading. Another image of the sage, this time standing with outstretched arms, was located in the Uzbek city of Urgench.

On September 6, 1983, the Soviet government released a postal series of a commemorative stamp featuring the face of the Persian sage, with the inscription “1200 years” in reference to the 1200 years of his probable birth. In 2012 the Uzbek government also released a commemorative postage stamp of Khuarismi, inspired by the statue of the sage that currently stands in Khiva.



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  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. ^ Victor J. Katz, STAGES IN THE HISTORY OF ALGEBRA WITH IMPLICATIONSFOR TEACHING (PDF), in VICTOR J.KATZ, University of the District of Columbia Washington DC, USA, p. 190. URL consultato il 7 ottobre 2017 (archiviato dall”url originale il 27 marzo 2019). Ospitato su 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. ^ (EN) John L. Esposito, The Oxford History of Islam, Oxford University Press, 6 aprile 2000, p. 188, ISBN 978-0-19-988041-6.«Al-Khwarizmi is often considered the founder of algebra, and his name gave rise to the term algorithm.»
  14. ^ Cfr. in tal senso M. Dunlop, “Muḥammad b. Mūsā al-Khwārizmī”, in Journal of the Royal Asiatic Society (1943), pp. 248-250.
  15. Gerald J. Toomer: «Al-Khwārizmī, Abū Ja’far Muhammad Ibn Mūsā» (Αγγλικά) Charles Scribner”s Sons. Δεκαετία του 1970.