Al-Khwarizmi
Muhammad ibn Musa al-Khwarizmi, or simply al-Khwarizmi was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in Baghdad, the contemporary capital city of the Abbasid Caliphate. One of the most prominent scholars of the period, his works were widely influential on later authors, both in the Islamic world and Europe.
His popularizing treatise on algebra, compiled between 813 and 833 as Al-Jabr, 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. Because al-Khwarizmi was the first person to treat algebra as an independent discipline and introduced the methods of "reduction" and "balancing", he has been described as the father or founder of algebra. The English term algebra comes from the short-hand title of his aforementioned treatise. His name gave rise to the English terms algorism and algorithm; the Spanish, Italian, and Portuguese terms algoritmo; and the Spanish term guarismo and Portuguese term algarismo, all meaning 'digit'.
In the 12th century, Latin translations of [|al-Khwarizmi's textbook on Indian arithmetic], which codified the various Indian numerals, introduced the decimal-based positional number system to the Western world. 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.
Al-Khwarizmi revised Geography, the 2nd-century Greek-language treatise by Ptolemy, listing the longitudes and latitudes of cities and localities. He further produced a set of astronomical tables and wrote about calendric works, as well as the astrolabe and the sundial. Al-Khwarizmi made important contributions to trigonometry, producing accurate sine and cosine tables.
Life
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. He was of Persian stock; his name means 'from Khwarazm', a region that was part of Greater Iran, and is now part of Turkmenistan and Uzbekistan.Al-Tabari gives his name as Muḥammad ibn Musá al-Khwārizmī al-Majūsī al-Quṭrubbullī. The epithet al-Qutrubbulli could indicate he might instead have come from Qutrubbul, near Baghdad. However, Roshdi Rashed denies this:
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.
Regarding al-Khwārizmī's religion, Toomer writes:
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. 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.
During the reign of al-Wathiq, he is said to have been involved in the first of two embassies to the Khazars. 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.
Contributions
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.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. When the work was translated into Latin in the 12th century as Algoritmi de numero Indorum, the term "algorithm" was introduced to the Western world.
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, presenting the coordinates of places based on those in the Geography of Ptolemy, but with improved values for the Mediterranean Sea, Asia, and Africa.
He wrote on mechanical devices like the astrolabe and sundial. He assisted a project to determine the circumference of the Earth and in making a world map for al-Ma'mun, the caliph, overseeing 70 geographers. 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.
Algebra
Al-Jabr 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. The term "algebra" is derived from the name of one of the basic operations with equations described in this book. The book was translated in Latin as Liber algebrae et almucabala by Robert of Chester 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.It provided an exhaustive account of solving polynomial equations up to the second degree, 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.
Al-Khwārizmī's method of solving linear and quadratic equations worked by first reducing the equation to one of six standard forms
- squares equal roots
- squares equal number
- roots equal number
- squares and roots equal number
- squares and number equal roots
- roots and number equal squares
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,
In modern notation this process, with x the "thing" or "root", is given by the steps,
Let the roots of the equation be x = p and x = q. Then, and
So a root is given by
Several authors have published texts under the name of, 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:
Victor J. Katz adds :
John J. O'Connor and Edmund F. Robertson wrote in the MacTutor History of Mathematics Archive:
Roshdi Rashed and Angela Armstrong write:
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. 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.
Arithmetic
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, and perhaps a more elementary text, kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī. These texts described algorithms on decimal numbers that could be carried out on a dust board. Called takht in Arabic, 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.As part of 12th century wave of Arabic science flowing into Europe via translations, these texts proved to be revolutionary in Europe. 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.
Four Latin texts providing adaptions of Al-Khwarizmi's methods have survived, even though none of them is believed to be a literal translation:
- Dixit Algorizmi
- Liber Alchoarismi de Practica Arismetice
- Liber Ysagogarum Alchorismi
- Liber Pulveris
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.