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.
Few biographical details are known about al-Khwarizmi's life. His popularizing treatise on algebra, compiled between 813 and 833 as Al-Jabr (The Compendious Book on Calculation by Completion and Balancing),Oaks, J. (2009), "Polynomials and Equations in Arabic Algebra", Archive for History of Exact Sciences, 63(2), 169–203. 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" (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),(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." he has been described as the fatherBoyer, 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..."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." 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 ; and the Spanish term and Portuguese term , 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.van der Waerden, Bartel Leendert (1985). A History of Algebra: From al–Khwarizmi to Emmy Noether. Berlin: Springer-Verlag. 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.
Al-Khwarizmi's name was latinized as Algoritmi, making his name the origin of the word "algorithm."
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."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. He was of Persian descent;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)." his name means 'from Khwarazm', a region that was part of Greater Iran,Encycloaedia Iranica-online, s.v. "CHORASMIA, ii. In Islamic times ," by Clifford E. Bosworth. and is now part of Turkmenistan and Uzbekistan. Despite being of non-Arabic origin (probably Khwārezmian), he wrote his scientific works entirely in Arabic.
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 (Qatrabbul),"Iraq After the Muslim Conquest", by Michael G. Morony, (a 2005 facsimile from the original 1984 book), p. 145 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 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.Maher, P. (1998), "From Al-Jabr to Algebra", Mathematics in School, 27(4), 14–15. 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 figures such as 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 825, 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 (Al-Khwarizmi on the Hindu art of reckoning), 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 ("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.
He wrote on mechanical devices like the astrolabeJoseph Frank, al-Khwarizmi über das Astrolab, 1922. 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.
Al-Khwarizmi's contributions to astronomy, which mixed theory and practice, were directly impactful upon astronomers of his epoch and the field as a whole. His meticulous methodological approach and scientific developments in this field directly influenced modern astronomy: "Al-Khwarizmi's astronomical tables, known as Zij, became essential tools for subsequent generations of astronomers. These tables enabled precise calculations for predicting astronomical events, such as eclipses and the positions of celestial bodies, which were crucial for navigation and timekeeping."
Algebra
Al-Jabr (The Compendious Book on Calculation by Completion and Balancing, ) 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 (, 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.
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. "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."
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 ( "restoring" or "completion") and ("balancing"). 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. 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 the 1831 "Rosen" translation)
In modern notation this process, with x the "thing" ( shayʾ) 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 :