al-Khwārizmī, Muslim mathematician and astronomer whose major works introduced Hindu-Arabic numerals and the concepts of algebra into European mathematics. Latinized versions of his name and of his most famous book title live on in the terms algorithm and algebra.
Few details of al-Khwārizmī’s life are known with certainty. Ibn al-Nadim gives his birthplace as Khwarazm, a region that was part of Greater Iran, and is now part of Turkmenistan, and Uzbekistan. After the Muslim conquest of Persia, Baghdad had become the centre of scientific studies and trade, and many merchants and scientists from as far as China and India traveled there, as did al-Khwārizmī. He worked in the House of Wisdom established by the Abbasid Caliph al-Ma’mūn, where he studied the sciences and mathematics and later he became director of this institution. The House of Wisdom acquired and translated scientific and philosophical treatises, particularly Greek and Indian mathematical and astronomy works (including those of Brahmagupta) into Arabic, and produced original work which had a lasting influence on the advance of Muslim and (after his works spread to Europe through Latin translations in the 12th Century) later European mathematics, as well as publishing original research. Al-Khwārizmī’s work on elementary algebra, Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr waʾl-muqābala (“The Compendious Book on Calculation by Completion and Balancing”), was translated into Latin in the 12th century, from which the title and term algebra derives. Algebra is a compilation of rules, together with demonstrations, for finding solutions of linear and quadratic equations based on intuitive geometric arguments, rather than the abstract notation now associated with the subject. Its systematic, demonstrative approach distinguishes it from earlier treatments of the subject. It also contains sections on calculating areas and volumes of geometric figures and on the use of algebra to solve inheritance problems according to proportions prescribed by Islamic law. Elements within the work can be traced from Babylonian mathematics of the early 2nd millennium BCE through Hellenistic, Hebrew, and Hindu treatises.
In the 12th century a second work by al-Khwārizmī introduced Hindu-Arabic numerals and their arithmetic to the West. It is preserved only in a Latin translation, Algoritmi de numero Indorum (“Al-Khwārizmī Concerning the Hindu Art of Reckoning”). From the name of the author, rendered in Latin as Algoritmi, originated the term algorithm.
A third major book was his Kitāb ṣūrat al-arḍ (“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 (flourished 127–145 CE), supervising some 70 geographers, he revised and expanded Egyptian polymath Ptolemy’s earlier work on geography to cover the coordinates of some 2400 places throughout the world, particularly around the Mediterranean Sea and cities in Africa and Asia, including lists with latitudes and longitudes, cities, seas, mountains, islands and rivers. He also assisted in the construction of a world map for al-Maʾmūn and participated in a project to determine the circumference of the Earth, which had long been known to be spherical, by measuring the length of a degree of a meridian through the plain of Sinjār in Iraq.
Finally, al-Khwārizmī also compiled a set of astronomical tables (Zīj), based on a variety of Hindu and Greek sources. This work included a table of sines, evidently for a circle of radius 150 units. Like his treatises on algebra and Hindu-Arabic numerals, this astronomical work (or an Andalusian revision thereof) was translated into Latin.
He also made important contributions to trigonometry, producing accurate sine and cosine tables, and the first table of tangents.