Islamic Mathematics in the Medieval Period

Overview

During the medieval period Islamic mathematicians enjoyed a dynamic and vibrant profession that, contrary to many popular teachings, made significant contributions to their field that continue to affect the way mathematics is practiced today. They did not simply preserve the glories of Greek mathematics and transfer some concepts from Hindu mathematicians to Europe. Rather, they developed sophisticated systems of algebra, introduced many now-standard mathematical notations, and helped mathematics move away from the largely geometrical formulations of the Greeks to a more symbolic and abstract structure that is much closer to the manner in which mathematics is practiced today.

Background

The first great flowerings of mathematics occurred in Babylonia, Egypt, and Greece. With the passing of time, these cultures either vanished or became assimilated into the Roman Empire. In particular, the Greek tradition in mathematics helped establish the form of European and Roman mathematics for several centuries.

To the east, China, Persia, and India were also developing their own mathematical traditions, largely independent of one another and wholly independent of Europe. With the fall of Rome in the fifth century, progress in mathematics largely stopped in Europe, although Asian cultures remained unaffected.

In the seventh century, in the Middle East, Muhammad founded the Islamic religion. Within a few decades, Islam had spread throughout the Arabian peninsula and, within a century, it became one of the dominant religions in the Middle East. Although followers of Islam did not originally express much of an interest in study, this quickly changed and they set about copying, translating, and adding to the mathematical knowledge of Greece, India, Persia, and (when possible) China.

It was long thought that Islamic scholars did little more than simply translate Greek texts, holding them until Europe emerged from its intellectual interregnum during the Middle Ages. In reality, Islamic mathematicians did far more than this. Specifically, they invented the algebra that most learn in school today, made significant advances in the field of trigonometry, and helped form a synthesis of mathematical ideas, fusing the best of Greek mathematics with important Hindu and Persian concepts to create a mathematical structure that was far grander than what they had inherited.

For several centuries, the Middle East, specifically Baghdad, was the world's center for mathematics. Islamic scholars such as al-Khwarizmi (780?-850?), Thabit ibn Qurra (836?-901), Abu al-Wafa (940-998), al-Kharki, al-Biruni (973-1048), Omar Khayyam (1048?-1131), and others literally reshaped the entire field, benefiting all of mathematics and much of science.

Impact

The most succinct way to describe the impact of Islamic mathematicians is to note that they completely changed the "flavor" of mathematics during their dominance in the field. One example of this is the change from the largely geometric formulations of the Greeks to the largely symbolic formulations that we use today. In other words, most of the problems addressed by Greek mathematicians were solved by describing them mathematically and then trying to find a geometric solution, such as the famous solution to the Pythagorean Theorem, which shows a triangle with a square drawn on each of the triangle's sides. In addition, the introduction of numerals, borrowed from Hindu culture, greatly facilitated the performing of calculations (try to imagine solving a problem that goes "first you have to find the product of twenty-one and fifteen" instead of simply reading 21 × 15 = ?). Islamic contributions to trigonometry also served to meld the Greek and Hindu approaches, and their contributions to number theory were equally impressive.

Of prime importance was the development of algebra (originally al-jabr) by the great Arab mathematician al-Khwarizmi (770-840). Simply put, algebra is a way of letting symbols represent numbers or other concepts, and manipulating those symbols according to a fixed and logical set of rules. Most learn aspects of al-Khwarizmi's algebra in school as they learn, for example, to solve an equation to find the value of x. Other algebraic problems include finding the "roots" of an equation, that is, where a particular line or curve will intersect the x-axis on a piece of graph paper. In general, any problem in which symbols are manipulated in order to arrive at a final answer is a type of algebra problem. Extracting a square root is yet another example of an algebraic problem.

This was a crucial step in the advancement of mathematics because it freed mathematicians from either performing interminable calculations for every problem solved or from having to try to construct a geometrical representation of the problem and trying to solve that. Although the Greeks brought geometrical mathematics to surprising levels of sophistication, this methodology is inherently limited because one can only solve problems that can be diagrammed in a way that lends itself to solution. The majority of mathematical problems are not amenable to solutions in this manner and, in any event, it is an indirect way to work in most cases. By abstracting the problem and using symbols to represent the important variables and concepts, it is possible to carry the problem to an analytical (or "symbolic") solution and, as the final step, to "plug in" actual numbers to find a numerical solution to the problem. This is the way that most scientific and engineering calculations are performed, and it is a powerful conceptual tool to be able to bring to bear on any problem.

The introduction of the Hindu numbering system was hardly less important to the practice of mathematics. By using a single symbol for each of the 10 numerals and using zero as a placeholder, mathematical calculations suddenly became almost simple. For example, consider the following problem, written in both Roman and Hindu notation:

CXXVI multiplied by XII = ?

126 × 12 = 252 + 1260 = 1512

In both cases, of course, the final answer is the same. However, solving this problem using Roman numerals is almost impossible; better to use an abacus or some other multiplication aid. In addition to aiding in computation, Hindu numerals were just easier and less ambiguous to read. One might erroneously read XIII as XII, for example, or might write XI as IX, errors far less likely to occur when each numeral is unique.

Islamic mathematicians seemed almost giddy with the possibilities raised by these two concepts. Throughout the mathematics of this era we find impossibly long tables filled with numbers, each the result of painstaking calculations. Some mathematicians actually completed these calculations in base 60 as well as in base 10, and one table contained nearly a half million entries. Although, by today's standards, this sort of work seems pointless (not to mention mind-numbingly dull), it was important to undertake at that time. Future mathematicians were able to use these tables of numbers in much the same way that, until the advent of cheap and powerful computers and calculators, statisticians used tables of statistical data, scientists used tables of logarithms and trigonometric functions, and so forth. By devoting their lives to calculating and completing these tables, Islamic mathematicians did a great service to their colleagues and to those who were to follow.

Islamic contributions to number theory are far from being the only other contribution to mathematicians during this intellectual golden age. Indeed, entire books have been written on this one subject. However, it is fair to say that Islamic contributions to number theory during the medieval period were also both important and impressive.

In some respects, it is difficult to distinguish between pure number theory at the level practiced by medieval Islamic mathematicians and the algebra they introduced. Simply put, number theory is the study of the properties of numbers. For example, the search for more digits in the number p, the search for a pattern among these digits, or the proof that no such pattern exists are examples of number theory. In a more practical area, number theory also deals with developing methods of extracting the roots of numbers (such as square or cube roots), or with finding pairs of numbers that meet certain criteria.

Number theory is often thought of as mathematics at its purest because, for the most part, discoveries in number theory have little practical utility and are sought instead for their mathematical and conceptual beauty. Although Islamic mathematicians did not originate this area of inquiry, they certainly pursued it vigorously, discovering many properties of prime numbers and several interesting and important theorems that would not be proven for several centuries, and asking questions that helped bring to number theory a degree of mathematical and intellectual rigor.

In general, it can be claimed that Islamic mathematics represented a sort of middle road between the mathematics of Greece and that of India, with a particularly Islamic flair. Islamic mathematicians translated virtually every surviving Greek text on mathematics and they were certainly aware of the Greek discoveries and formulations of problems. In fact, the earliest Muslim text describing algebra describes problems that could only have been translated from the Greeks. To this, they added the numbering system used by mathematicians in the East, as well as some Hindu mathematical techniques. However, their approach was more analytical than that of the Greeks and less mystical than that of the Hindus.

The golden age of Islamic mathematics lasted for only a few centuries, from about the eighth century through the twelfth century. There were several subsequent resurgences, and Islamic mathematics did not go into a decline until about the fifteenth century, with the death of al-Kashi (1380?-1429), the last great Islamic mathematician of this era. By that time, however, Europe was recovering from its long intellectual stagnation, ready to accept back the mathematics it had ignored for so long.

P. ANDREW KARAM