Al-Khwarizmi

Al-Khwarizmi the ninth-century Persian mathematician and astronomer whose name, Latinised as Algoritmi, gave the world the word algorithm and whose treatise on Hindu numerals introduced the decimal system to the Islamic world and then to Europe. His book on algebra founded a new mathematical discipline.

Life and Setting

Muhammad ibn Musa al-Khwarizmi was born around 780 in the region of Khwarazm, south of the Aral Sea, from which he took his name, and he worked for most of his life in Baghdad under the early Abbasid caliphs. He is one of the most celebrated scholars of the court of al-Mamun, a ruler who gathered translators, physicians and scientists from across the empire.

He became a member of the House of Wisdom, the caliph's library and research institution, where Greek, Persian and Indian works of learning were collected and translated into Arabic. This extraordinary institution, a meeting point of civilisations, gave him access to the mathematical traditions of Egypt, Babylon, India and Greece.

His work combined the practical needs of the empire with genuine theoretical advance. He was concerned with astronomy for navigation and timekeeping, with arithmetic for trade and administration, and with geometry and algebra for the resolution of legal and commercial problems.

The Book of Algebra

Around 830 he composed his most famous work, the Compendious Book on Calculation by Completion and Balancing, which introduced the discipline the world now calls algebra. Its title supplied the term: al-jabr refers to the operation of restoring a term to the other side of an equation, and al-muqabala to the balancing of like quantities on the two sides.

The book presented systematic methods for solving linear and quadratic equations through completion and balancing, and it classified equations into cases according to their terms. It demonstrated the methods both for practical problems of trade, inheritance and surveying and, in its final part, for problems of pure mathematics requiring the geometry of Euclid and Apollonius.

Although earlier traditions had solved specific equations, al-Khwarizmi's treatment was the first to organise the subject as a general and teachable discipline, giving it rules, worked examples and demonstrations. For this reason he has been called the father of algebra, and the book remained a model for Arabic and Latin algebraists for several centuries.

Hindu Numerals and Arithmetic

His treatise on arithmetic, known only through the Latin translation made centuries later, explained the use of the Hindu-Arabic numerals and the place-value system of decimal calculation. He described the digits, the principle of position, and the techniques of addition, subtraction, multiplication, division, and even operations with fractions and the extraction of square roots.

The methods he recorded had their origins in India, but it was through his exposition that they entered the Arabic learned tradition and passed from there to Europe. When the Latin version of his book circulated in the twelfth century, it introduced European merchants and scholars to the convenience of calculating with nine symbols and a zero.

Because his name was attached to the techniques, the medieval Latin term algorism, describing calculation with the new numerals, became the ancestor of the modern word algorithm. His arithmetic thus left its mark on the very vocabulary of computing.

Geography and Astronomy

Al-Khwarizmi also worked extensively in astronomy, producing tables for the motions of the sun, the moon and the planets, and revising the astronomical tables of the Indian tradition in the light of Ptolemy. His zij was used in the Islamic world and in the West, where Latin adaptations circulated under his name well into the Middle Ages.

He took part in the project of determining the circumference of the earth and in the preparation of a geographical work describing the known world, its cities, rivers, mountains and seas. This survey, ordered by the caliph, gathered data from travelling merchants and officials and represents an early attempt at systematic world geography.

His instruments and methods, and his careful reduction of astronomical and geographical data, extended the practical mathematics of the age. Calendars, prayer times and the direction of Mecca all depended on them, and his tables helped fix the length of the year and the times of eclipses.

Influence and Legacy

Al-Khwarizmi's mathematics reached Europe by several routes, most notably through the translations and the Arithmetic of Leonardo of Pisa. From the twelfth century onward, his algebra and his numerals transformed European commerce, science and calculation, displacing the cumbersome Roman system and the abacus.

His authority persisted for centuries. Latin writers treated him as the model algebraist and the source of the Indian art of reckoning, and his name passed into common speech as the word for any step-by-step procedure. The very phrase used to describe a computational routine preserves his memory.

He is remembered as one of the founders of algebra and as a key transmitter of the decimal system, and he stands among the first great scientists of the Islamic golden age. Few scholars have had so direct an imprint on both the language and the methods of modern mathematics and computing.

See also

• Algebra
• House of Wisdom
• Hindu-Arabic numerals
• Algorithm