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 * Al-Khwarizmi redirects here, for other meanings see Al-Khwarizmi (disambiguation)

Muhammad ibn Musa al-Khwarizmi (Arabic: محمد بن موسى الخوارزمي Muḥammad ibn Mūsā al-Ḵwārizmī) was a mathematician, astronomer, astrologer and geographer of Persian origin. He was born around 780, in either Khwarizm or Baghdad, and died around 850.

He was the author of al-Kitāb al-muḵtaṣar fī ḥisāb al-ǧabr wa-l-muqābala, the first book on the systematic solution of linear and quadratic equations. Consequently he is considered to be the father of algebra, a title he shares with Diophantus. The word algebra is derived from al-ǧabr, one of the two operations used to solve quadratic equations, as described in his book. Algoritmi de numero Indorum, the Latin translation of his other major work on the Indian numerals, introduced the positional number system and the number zero to the Western world in the 12th century. The words algorism and algorithm stem from Algoritmi, the Latinization of his name.

Biography
Few details about his life are known: it is not even certain where al-Khwarizmi was born. His name indicates he might have came from Khwarizm in the Central Asia in Uzbekistan. It is now known as Khiva and located in Uzbekistan. The historian al-Tabari gave him the epithet al-Qutrubbulli, indicating that he might instead have came from Qutrubbull, a small town near Baghdad. Al-Tabari also gave him the epithet al-Majusi, suggesting that he was a Magus or Magi, a Zoroastrian priest. Some have suggested that the preface to his Algebra could be interpreted to indicate that he was an orthodox Muslim, though his practice of astrology does not support this hypothesis.

Al-Khwarizmi accomplished most of his work in the period between 813 and 833. After the Islamic conquest of Persia, Baghdad became the centre of scientific studies and trade, and many merchants and scientists, from as far as China and India traveled to this city--as such apparently so did Al-Khwarizmi. He worked in Baghdad as a scholar at the House of Wisdom established by Caliph al-Maʾmūn, where he studied and translated Greek scientific manuscripts.

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.

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 and Babylonian 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".

When his work was copied and transferred to Europe through Latin translations, it had a profound impact on the advancement of basic mathematics in Europe. He also wrote on mechanical devices like the astrolabe and sundial.

Algebra
al-Kitāb al-muḵtaṣar fī ḥisāb al-ǧabr wa-l-muqābala (Arabic: الكتاب المختصر في حساب الجبر والمقابلة “The Compendious Book on Calculation by Completion and Balancing”) is a mathematical book written approximately 820 AD.

The book extended the work of Indian mathematician Brahmagupta and Greek mathematician Diophantus on algebraic equations. The book is considered to have defined algebra. The word algebra is derived from the name of one of the basic operations with equations (al-jabr) described in this book. The book was translated in Latin as Liber algebrae et almucabala by Robert of Chester in 1145, hence "algebra".

Al-Khwarizmi's method of solving linear and quadratic equations worked by first reducing the equation to one of six standard forms (where a, 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)

using the two operations al-ǧabr ("completion") and al-muqābala ("balancing"). Al-ǧabr is the process of removing negative units, roots and squares from the equation by adding the same quantitiy 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.

Arithmetic
Algoritmi de numero Indorum ("al-Khwarizmi on the Hindu Art of Reckoning") on Arithmetic, which survived in a Latin translation but was lost in the original Arabic. The translation was most likely done in the 12th century by Adelard of Bath, who had also translated the astronomical tables in 1126. The original Arabic title was possibly Kitāb al-Ğamʿ wa-t-tafrīq bi-ḥisāb al-Hind.

Geography
Al-Khwarizmi's third major work is his Kitāb ṣūrat al-Arḍ (Arabic: كتاب صورة الأرض "Book on the appearance of the Earth" or "The image of the Earth" translated as Geography), which was finished in 833. It is a revised and completed version of Ptolemy's Geography, consisting of a list of 2402 coordinates of cities and other geographical features following a general introduction.

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. The complete title translates as Book of the appearance of the Earth, with its cities, mountains, seas, all the islands and rivers, written by Abu Ja'far Muhammad ibn Musa al-Khwarizmi, according to the geographical treatise written by Ptolemy the Claudian.

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 points out, this excellent system allows us to deduce many latitudes and longitudes where the only document in our possession 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 transfered 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.

One of the corrections which al-Khwarizmi made in Ptolemy's work is the reduction of the latitude of the Mediterranean from 62° to 52° when, in actual fact, it should be only 42°. The Arab opts for the same zero meridian as Ptolemy, that of the Canaries. The amount of inhabited land extends over 180°.

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 Atlantic coast of the Dragon's Tail, which does not exist in Ptolemy's map, is traced in very little detail on al-Khwarizmi's map, but is clear and precise on the Martellus map and on the later Behaim version.

The only real difference is that, on the Ptolemy map, the South American coast to the south of Cattigara curves west to join the African coast, whereas on the other maps, it curves to the east, north-east and north to form the great Dragon's Tail peninsula, that is to say, South America. This peninsula also exists on al-Khwarizmi's map, as we can see most clearly on Daunicht's reconstruction. As Gallez points out, with the identification of rivers and mountains, we have proved that on the Martellus map this peninsula is in actual fact South America, complete with both the Pacific and Atlantic coasts. After studying the Strait of Magellan on the al-Khwarizmi map as well as that of Martellus, we can see that Tierra del Fuego was better known in Baghdad in 833 than in Florence in 1489.

Astronomy
Al-Khwarizmi's last major work is his Zīğ or astronomical tables, which was based on a number of Greek and Indian sources. It included a table a sine values.

Other works
Al-Khwarizmi has written several other works including Risāla fi stiḫrāğ ta’rīḫ al-Yahūd on the Jewish calendar and a work on the astrolabe. Ibn an-Nadīm mentions Kitāb ar-Ruḫāma (Kitab al-Rukhmat), on the sundial, but this has been lost.