Al-Khwarizmi

Muhammad ibn Musa al-Khwarizmi was a Persian mathematician, astronomer and geographer who worked in Baghdad in the early ninth century, during the Islamic Golden Age. His book on solving equations gave the world the word algebra, his book on calculating with Indian numerals spread the decimal place-value system that is now used everywhere, and the Latin form of his own name became the word algorithm.

Very little is known about his life, and his dates, roughly 780 to 850, are estimates. His name indicates a family connection with Khwarazm, the oasis region on the lower Amu Darya river south of the Aral Sea, today divided between Uzbekistan and Turkmenistan. The tenth-century bibliographer Ibn al-Nadim says he came from there, and his first language was probably Khwarazmian, an Iranian language, though he wrote all his works in Arabic, the language of learning across the caliphate.

The historian al-Tabari gives him two further epithets. One, al-Majusi, means Zoroastrian, and has suggested to some that his family had been Zoroastrian. The pious Islamic preface to his algebra shows that he himself was a Muslim, so the label may refer to his ancestry. The other, al-Qutrubbulli, may link him to Qutrubbul, a district near Baghdad, although some scholars think it is a copyist's error.

Al-Khwarizmi's career belongs to Baghdad, the capital of the Abbasid Caliphate, under the caliph al-Ma'mun, who ruled from 813 to 833. Al-Ma'mun, a son of Harun al-Rashid, was a keen patron of science and supported a circle of scholars associated with the Bayt al-Hikma, the House of Wisdom, a royal library where Greek, Persian and Indian works were collected and translated into Arabic. Al-Khwarizmi worked there alongside scholars such as the Banu Musa brothers, and dedicated at least two of his books to the caliph. The scale and organisation of the House of Wisdom are debated by historians, but it is clear that Baghdad in these decades was the leading centre of scientific work in the world.

He also took part in al-Ma'mun's scientific projects, which included a world map and an effort to measure the size of the earth. One account says he was later sent on a mission to the Khazars under the caliph al-Wathiq, but this is uncertain.

His most famous work, written around 820, is usually known as The Compendious Book on Calculation by Completion and Balancing, in Arabic al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala. The two terms in the title name the two basic operations he used. Al-jabr, restoration or completion, means getting rid of a subtracted quantity by adding it to both sides of an equation. Al-muqabala, balancing, means cancelling equal quantities that appear on both sides.

Using these operations he showed that every linear or quadratic equation, the kinds involving a number, an unknown and its square, could be reduced to one of six standard forms, and he gave a method for solving each. There were no symbols: everything was written in words, with the unknown called the root and its square the wealth. A classic example from the book asks for a square which, together with ten of its roots, equals thirty-nine, in modern terms x² + 10x = 39. He solves it by halving the ten to get five, squaring that to get twenty-five, adding it to thirty-nine to make sixty-four, taking the square root, eight, and subtracting the five, which gives three. He then justified the rule with a geometric diagram, building a larger square out of a square and rectangles, the method now called completing the square. He recognised only positive solutions, as was usual at the time.

The second half of the book applies these techniques to practical matters, above all the complicated division of inheritances under Islamic law, as well as trade and the measuring of land. What made the work new was not any single result, since Babylonian, Greek and Indian mathematicians had solved similar problems, but the treatment of algebra as a subject in its own right, with general methods rather than collections of puzzles. Historians debate whether he or the Greek Diophantus deserves the title father of algebra; Diophantus used more abbreviations and harder problems, but al-Khwarizmi's systematic, elementary approach is much closer to the algebra later taught in schools.

A second book explained how to calculate with the nine Indian digits and a sign for zero, using place value so that the position of a digit shows whether it counts units, tens or hundreds. It set out methods for addition, subtraction, multiplication, division and finding square roots. The Arabic original is lost, but Latin versions survive, one beginning with the words Dixit Algorizmi, "al-Khwarizmi said". Medieval Europeans called the new methods algorism after him, and the word algorithm, now meaning any step-by-step procedure, comes from the same source. Because he presented the numerals to the Arabic world and his book later carried them to Europe, they became known in the West as Arabic numerals, although they originated in India.

His astronomical tables, the Zij al-Sindhind, drew heavily on Indian astronomy, especially the work of Brahmagupta, which had reached Baghdad a few decades earlier. They gave methods for calculating the positions of the sun, moon and planets, for eclipses and for calendars, together with tables of sines. The original is lost, but a revision made in al-Andalus was translated into Latin in the twelfth century, probably by Adelard of Bath. He also wrote on the astrolabe, the sundial and the Jewish calendar.

His Book of the Image of the Earth, completed in 833, reworked Ptolemy's Geography. It lists the latitudes and longitudes of some 2,400 places, arranged by climate zone. He corrected Ptolemy's exaggerated length of the Mediterranean and gave much better data for the Islamic lands, though for Europe he largely followed his Greek source.

In the twelfth century, scholars in Spain, where Christian, Muslim and Jewish learning met, translated his work into Latin. Robert of Chester translated the algebra at Segovia in 1145, and Gerard of Cremona made another version at Toledo. His algebra was used as a university textbook in Europe until the sixteenth century, and his arithmetic was a main source for the spread of Hindu-Arabic numerals, alongside the Liber Abaci of Leonardo of Pisa, known as Fibonacci, published in 1202. Later Arabic mathematicians, including Abu Kamil and Omar Khayyam, built directly on his algebra. Modern historians differ on how original he was: some have judged his science derivative, while others stress how much of later mathematics in both East and West grew from his clear, practical presentation. Algebra and algorithm, two words used daily in mathematics and computing, preserve his memory.