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Abū ‘Abd Allāh Muḥammad ibn Mūsā al-Khwārizmī (Arabic: أبو عبد الله محمد بن موسى الخوارزمي) was a Persian mathematician, astronomer, astrologer, geographer and author. Fairly little it known about his life but he was born around 780, most likely in Baghdad, and died around 850.

The word algebra is derived from the title of one of his books al-Kitāb al-mukhtasar fi hisab al-jabr wa’l-muqābalah (“The Compendious Book on Calculation by Completion and Balancing”) and consequently he is considered to be the father of algebra. The word algorithm is derived from his name.

Introduction

Khwarizmi was born in the town of Khwarizm (now Khiva), in Khorasan province of Persia (now in Uzbekistan). Some historians argue though that he was born in Qutrubulli, a small town near Baghdad as mentioned in the works of al-Tabari, however, it is generally considered he was born in Khiva/Khwarizm, hence his name. He accomplished most of his work in the period between 813 and 833. Mathematical historian Gandz gives this opinion of Khwarizmi's algebra:

"Khwarizmi's algebra is regarded as the foundation and cornerstone of the sciences. In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because 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." (1)

and Mohammad Khan, says:

"In the foremost rank of mathematicians of all time stands Khwarizmi. He composed the oldest works on arithmetic and algebra. They were the principal source of mathematical knowledge for centuries to come in the East and the West. The work on arithmetic first introduced the Hindu numbers to Europe, as the very name algorism signifies; and the work on algebra ... gave the name to this important branch of mathematics in the European world..."(2)

There is a widely held view that, after a brilliant period for mathematics when the Greeks laid the foundations for modern mathematics, there was a period of stagnation before the Europeans took over where the Greeks left off at the beginning of the sixteenth century. The common perception of the period of 1000 years or so between the ancient Greeks and the European Renaissance is that little happened in the world of mathematics except that some Arabic translations of Greek texts were made which preserved the Greek learning so that it was available to the Europeans at the beginning of the sixteenth century.

That such views should be generally held is of no surprise. Many leading historians of mathematics have contributed to the perception by either omitting any mention of Arabic/Islamic mathematics in the historical development of the subject or with statements such as that made by Duhem in :-

... Arabic science only reproduced the teachings received from Greek science.

The regions from which the "Arab mathematicians" came was centred on Iran/Iraq but varied with military conquest during the period. At its greatest extent it stretched to the west through Turkey and North Africa to include most of Spain, and to the east as far as the borders of China.

The background to the mathematical developments which began in Baghdad around 800 is not well understood. Certainly there was an important influence which came from the Hindu mathematicians whose earlier development of the decimal system and numerals was important. There began a remarkable period of mathematical progress with al-Khwarizmi's work and the translations of Greek texts.

This period begins under the Caliph Harun al-Rashid, the fifth Caliph of the Abbasid dynasty, whose reign began in 786. He encouraged scholarship and the first translations of Greek texts into Arabic, such as Euclid's Elements by al-Hajjaj, were made during al-Rashid's reign. The next Caliph, al-Ma'mun, encouraged learning even more strongly than his father al-Rashid, and he set up the House of Wisdom in Baghdad which became the centre for both the work of translating and of of research. Al-Kindi (born 801) and the three Banu Musa brothers worked there, as did the famous translator Hunayn ibn Ishaq.

It should be emphasised that the translations into Arabic at this time were made by scientists and mathematicians such as those named above, not by language experts ignorant of mathematics, and the need for the translations was stimulated by the most advanced research of the time. It is important to realise that the translating was not done for its own sake, but was done as part of the current research effort.

Contributions

He made major contributions to the fields of algebra, trigonometry, astronomy/astrology, geography and cartography. His systematic and logical approach to solving linear and quadratic equations gave shape to the discipline of algebra, a word that is derived from the name of his 830 book on the subject, al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala (الكتاب المختصر في حساب الجبر والمقابلة) or: "The Compendious Book on Calculation by Completion and Balancing". The book was first translated into Latin in the twelfth century.

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. As Rashed writes in (see also ):-

Al-Khwarizmi's successors undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose.

His book On the Calculation with Hindu Numerals written about 825, was principally responsible for the diffusion of the Indian system of numeration in the Middle-East and then Europe. This book also translated into Latin in the twelfth century, as Algoritmi de numero Indorum. From the name of the author, rendered in Latin as algoritmi, originated the term algorithm.

Some of his contributions were based on earlier Persian Astronomy Indian numbers and Greek sources.

Al-Khwarizmi systematized and corrected Ptolemy's data in geography as regards to Africa and the Middle east. Another major book was his Kitab surat al-ard ("The Image of the Earth"; translated as Geography), which presented the coordinates of localities in the known world based, ultimately, on those in the Geography of Ptolemy but with improved values for the length of the Mediterranean Sea and the location of cities in Asia and Africa.

He also assisted in the construction of a world map for the caliph al-Ma'mun and participated in a project to determine the circumference of the Earth, supervising the work of 70 geographers to create the map of the then "known world".(3)

When his work was copied and transferred to Europe through Latin translations, it made a profounding impact on the advancement of basic mathematics in Europe. He also wrote on mechanical devices like the clock, astrolabe, and sundial. His other contributions include tables of trigonometric functions, refinements in the geometric representation of conic sections, and aspects of the calculus of two errors.

Famous works

Al-Jabr wa-al-Muqabilah (literally, Calulating by Completion and Balancing ) from whose title came the name "Algebra"
Kitab al-Jam'a wal-Tafreeq bil Hisab al-Hindi (on Arithmetic, which survived in a Latin translation but was lost in the original Arabic)
Kitab Surat-al-Ard (on geography)
Istikhraj Tarikh al-Yahud (about the Jewish calendar)
Kitab al-Tarikh (literally, the book of history)
Kitab al-Rukhmat (about sun-dials)

Notes

“Father of Abdullah, Mohammed, son of Moses, native of Khwārizm”. Many alternative translations of his name exist: Abu .

References

S Gandz, The sources of al-Khwarizmi's algebra, Osiris, i (1936), 263-77.
A A al'Daffa, The Muslim contribution to mathematics (London, 1978).
From his biography in Encyclopædia Britannica.

Other sources to use

Books:

Biography in Dictionary of Scientific Biography (New York 1970-1990).
J N Crossley, The emergence of number (Singapore, 1980).
A F Faizullaev, The scientific heritage of Muhammad al-Khwarizmi (Russian) (Tashkent, 1983).
S Gandz (ed.), The geometry of al-Khwarizmi (Berlin, 1932).
E Grant (ed.), A source book in medieval science (Cambridge, 1974).
O Neugebauer, The exact sciences in Antiquity (New York, 1969).
R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994).
R Rashed, Entre arithmétique et algèbre: Recherches sur l'histoire des mathématiques arabes (Paris, 1984).
F Rosen (trs.), Muhammad ibn Musa Al-Khwarizmi : Algebra (London, 1831).

Articles:

K F Abdulla-Zade, al-Khwarizmi and the Baghdad astronomers (Russian), in The great medieval scientist al-Khwarizmi (Tashkent, 1985), 178-183.
M Abdullaev, al-Khwarizmi and scientific thought in Daghestan (Russian), in The great medieval scientist al-Khwarizmi (Tashkent, 1985), 228-232.
A Abdurakhmanov, al-Khwarizmi : great mathematician (Russian), in The great medieval scientist al-Khwarizmi (Tashkent, 1985), 149-151.
M A Akhadova, The mathematics of India and al-Khwarizmi (Russian), in The great medieval scientist al-Khwarizmi (Tashkent, 1985), 238-240.
S al-Khalidi, al-Khwarizmi : scholar of astronomical and mathematical geography (Arabic), in Proceedings of the Seventh Annual Conference on the History of Arabic Science (Arabic) (Aleppo, 1986), 55-63.
A Allard, La diffusion en occident des premières oeuvres latines issues de l'arithmétique perdue d'al-Khwarizmi, J. Hist. Arabic Sci. 9 (1-2) (1991), 101-105.
P G Bulgakov, al-Biruni and al-Khwarizmi (Russian), in Mathematics and astronomy in the works of scientists of the medieval East (Tashkent, 1977), 117-122, 140.
J N Crossley and A S Henry, Thus spake al-Khwarizmi : a translation of the text of Cambridge University Library ms. Ii.vi.5, Historia Math. 17 (2) (1990), 103-131.
Y Dold-Samplonius, Developments in the solution to the equation cxÛ + bx = a from al-Khwarizmi to Fibonacci, in From deferent to equant (New York, 1987), 71-87.
R Z Du, al-Khwarizmi and his algebraic treatise (Chinese), Math. Practice Theory (1) (1987), 79-85.
K Fogel, How al-Khwarizmi became known in Germany (Russian), in The great medieval scientist al-Khwarizmi (Tashkent, 1985), 85-91.
J P Hogendijk, al-Khwarizmi's table of the "sine of the hours" and the underlying sine table, Historia Sci. 42 (1991), 1-12.
B B Hughes, Robert of Chester's Latin translation of al-Khwarizmi's 'al-Jabr', Boethius : Texts and Essays on the History of the Exact Sciences XIV (Stuttgart, 1989).
D K Ibadov, The work of al-Khwarizmi in the estimation of Eastern encyclopedic scholars of the 10th - 16th centuries (Russian), in The great medieval scientist al-Khwarizmi (Tashkent, 1985), 265-268.
W Kaunzner, Über eine frühe lateinische Bearbeitung der Algebra al-Khwarizmis in MS Lyell 52 der Bodleian Library Oxford, Arch. Hist. Exact Sci. 32 (1) (1985), 1-16.
E S Kennedy, Al-Khwarizmi on the Jewish calendar, Scripta Math. 27 (1964), 55-59.
A S Kennedy and W Ukashah, al-Khwarizmi's planetary latitude tables, Centaurus 14 (1969), 86-96.
M M Khairullaev, al-Khwarizmi and his era (Russian), Voprosy Istor. Estestvoznan. i Tekhn. (3) (1983), 121-127.
P Kunitzsch, al-Khwarizmi as a source for the 'Sententie astrolabii', in From deferent to equant (New York, 1987), 227-236.
G P Matvievskaya, The algebraic treatise of al-Khwarizmi (Russian), in On the history of medieval Eastern mathematics and astronomy (Tashkent, 1983), 3-22.
G P Matvievskaya, History of the study of the scientific work of al-Khwarizmi (Russian),, in The great medieval scientist al-Khwarizmi (Tashkent, 1985), 72-82.
C A Nallino, Al'Khuwarizimi e il suo rifacimento della Geografia di Tolomeo, Raccolta di scritti editie inediti V (Rome, 1944), 458-532.
K H Parshall, The art of algebra from al-Khwarizmi to Viète : a study in the natural selection of ideas, Hist. of Sci. 26 (72, 2) (1988), 129-164.
M A Pathan, Al-Khwarizmi, Math. Ed. 6 (2) (1989), 92-94.
D Pingree, al-Khwarizmi in Samaria, Arch. Internat. Hist. Sci. 33 (110) (1983), 15-21.
B A Rosenfeld, 'Geometric trigonometry' in treatises of al-Khwarizmi, al-Mahani and ibn al-Haytham, in Vestigia mathematica (Amsterdam, 1993), 305-308.
B A Rozenfeld, al-Khwarizmi's spherical trigonometry (Russian), Istor.-Mat. Issled. 32-33 (1990), 325-339.
B A Rozenfeld, Number theory, geometry and astronomy in al-Khwarizmi's 'Book of Indian arithmetic' (Russian), in The great medieval scientist al-Khwarizmi (Tashkent, 1985), 66-72.
B A Rozenfeld and N D Sergeeva, The astronomical treatises of al-Khwarizmi (Russian), Istor.-Astronom. Issled. 13 (1977), 201-218.
M M Rozhanskaya, The historical-astronomical value of al-Khwarizmi's "zij" (Russian), in The great medieval scientist al-Khwarizmi (Tashkent, 1985), 158-165.
A S Sadykov, al-Khwarizmi : his era, life and work (Russian), in The great medieval scientist al-Khwarizmi (Tashkent, 1985), 8-13.
M Sani, The life and work of al-Khwarizmi, Menemui Mat. 4 (1) (1982), 1-9.
K S Siddikov, Muhammad al-Khwarizmi : creator of algebra (Russian), in The great medieval scientist al-Khwarizmi (Tashkent, 1985), 152-154.
S Kh Sirazhdinov and G P Matvievskaya, Muhammad ibn Musa al-Khwarizmi and his contribution to the history of science (Russian), Voprosy Istor. Estestvoznan. i Tekhn. (1) (1983), 108-119.
Z K Sokolovskaya, The "pretelescopic" period of the history of astronomical instruments. al-Khwarizmi in the development of precision instruments in the Near and Middle East (Russian), in The great medieval scientist al-Khwarizmi (Tashkent, 1985), 165-178.
B van Dalen, Al'Khwarizmi's astronomical tables revisited : analysis of the equation of time, in From Baghdad to Barcelona (Barcelona, 1996), 195-252.
K Vogel, Wie wurden al-Khwarizmi s mathematische Schriften in Deutschland bekannt?, Sudhoffs Arch. 68 (2) (1984), 230-234.
A I Volodarskii, al-Khwarizmi and Indian mathematics (Russian), in The great medieval scientist al-Khwarizmi (Tashkent, 1985), 232-238.
E Yu Yusupov and M M Kharullaev, The creative legacy of al-Khwarizmi and his place in the history of science (Russian), Voprosy Filos. (8) (1983), 140-146, 174.
Kh Zemanek, Manuscripts of al-Khwarizmi's works (Russian), in The great medieval scientist al-Khwarizmi (Tashkent, 1985), 115-121.
V K Zharov, Instrumental counting in al-Khwarizmi's arithmetical treatise (Russian), in The great medieval scientist al-Khwarizmi (Tashkent, 1985), 154-157.

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