VI.5 Abu Ja’far Muhammad ibn Ms al-Khwrizm

b. Unknown, 800; d. Unknown, 847 Arithmetic; algebra

Al-Khwrizm, or possibly his ancestors, came from Khwrizm (the modern region of Khorezm in Uzbekistan, also known as Khiva). Most of his life was spent as a scholar at the House of Wisdom, Baghdad, where he produced works on astronomy, mathematics, and geography. Of his mathematical works, two have come down to us, one on arithmetic and one on algebra.

The arithmetical work, which did not survive in Arabic and is known only through Latin translations, was the means by which Hindu numerals were transmitted to the West, as well as the corresponding methods of arithmetical calculation. Although the text was clearly based on Indian writings, in Europe the techniques became particularly associated with al-Khwrizm’s name in the form of algorism (from which the modern term “algorithm” is derived).

Al-Khwrizm’s al-Kitb al-mukhtaar f isb al-jabr wa’l-muqbala (“The compendious book on calculation by completion and balancing”) became the starting point for the subject of algebra for Islamic mathematicians. A work of elementary practical mathematics, it is written in three parts: one was devoted to solving equations, one to practical mensuration (areas and volumes), and one to problems that arose mainly from the complicated Islamic laws of inheritance (involving arithmetic and simple linear equations). No algebraic symbolism is employed: everything, including numerals, is expressed in words. The text opens with a brief discussion of the place-value system and then deals with equations of the first and second degrees. Remarkably, al-Khwrizm did not regard these equations just as a means for solving problems, as his predecessors had done, but studied them in their own right, classifying them into six separate types. In modern notation these are

ax² = bx,    ax² = b,    ax = b,
ax² + bx = c,    ax² + c = bx,    ax² = bx + c,

where a, b, and c are positive integers. The different types are necessary because al-Khwrizm did not recognize the existence of either negative numbers or zero as coefficients. Not only did al-Khwrizm give proofs that his methods worked, which in itself was not standard at the time, but the proofs he gave were geometrical ones. That is, they were not classical Greek proofs but geometrical demonstrations of the validity of his methods.

The key word of the Arabic title, al-jabr (“completion” or “restoration”), which refers to restoring all the terms to a standard form, eventually came into common usage in the West as algebra. It is, however, doubtful that al-Khwrizm’s work was the first Islamic work bearing that name.

Further Reading

Berggren, J. L. 1986. Episodes in the Mathematics of Medieval Islam. New York: Springer.