Islamic Mathematics

Summary: Arabic and Islamic mathematicians popularized the decimal system and Arabic numerals and also developed algebra.

Mathematicians living in Islamic lands and writing in Arabic have played a central role in the development of mathematics, particularly during the 700-year period from around the year 750 c.e to around 1450 c.e. These scholars lived in an area that not only includes the present-day Middle East but stretches into the western parts of India, the major cities of central Asia, all of northern Africa, and most of the Iberian Peninsula. Most of the influential mathematicians of this seventh-century era were Muslim, and most wrote in Arabic. However, the lands ruled by Muslim rulers included many ethnicities, cultures, languages, and religions. Muslims, Christians, Jews, Zoroastrians, Manichaeans, Sabians, Buddhists, Hindus, Persians, Turks, Sogdians, Mongols, Arabs, Berbers, Egyptians, and many others contributed to a remarkable multiethnic, multicultural civilization. Mathematics was not an exception. The full story of mathematics in this era has yet to be told. Hundreds of manuscripts await examination, translation, and a critical edition. Undoubtedly, in the years to come, our understanding of the extent, the import, and the influence of the mathematics of this period will change dramatically.

While their knowledge of what came before them was incomplete and uneven, the mathematicians of the Islamic era were aware of—and in some ways heirs to—ideas, methods, and points of view that originated in India, Persia, and—especially—Greek Alexandria. A remarkable translation movement coupled with a scholarly tradition of writing commentaries on previous works meant that mathematicians of this era were comfortable with the contents and the methodology of the works of, among others, Euclid, Archimedes, Apollonius, Ptolemy, and Diophantus as well as the basics of Indian decimal arithmetic and trigonometry. They also had access to Persian astronomical tables. They accomplished a great deal with this heritage. What the mathematicians of the Islamic era bequeathed to those who came later was very different in content, style, and approach than what had come before them. (A note on names: names of mathematicians and places can be transliterated to English based on their Arabic, Persian, or Turkish versions. For the most part, we have chosen what is currently most common in English. The one exception is that we have often omitted the Arabic definite article “al” that precedes titles and nicknames.)

The Decimal System and the Concept of Number

For Euclid—the preeminent mathematician of Greek Alexandria—“number” meant a rational number. In his work, irrational numbers were called magnitudes and were treated quite differently from numbers. In fact, Euclid’s very influential book Elements contains few numbers and hardly any calculations. Starting with Khwarizmi of Khwarizm (c. 780–850 c.e.), the principles of the positional decimal system that had originally come from India were organized and widely disseminated. Hence, with the use of 10 symbols it was possible to carry out all arithmetic operations. Over the following centuries, the methods for these arithmetical operations were improved and included working with decimal fractions and with large numbers. In fact, in the process, the Euclidean concept of number was gradually enlarged to include irrational numbers and their representation as decimal fractions. The mathematician Kashani (c. 1380–1429), also known as al-Kashi, worked comfortably with irrational numbers and, for example, was able to produce an approximation that was correct to 16 decimal places. The Arabic texts on the decimal number system were translated to Latin and were the basis for what are now called the Hindu-Arabic numerals.

Algebra

While it is possible to recognize algebraic problems in ancient mathematics, algebra as a discipline distinct from geometry and concerned with solving of equations was developed during the Islamic period. The first book devoted to the subject was Khwarizmi’s Al-kitab al-muhtasar fi hisab al-jabr wa-l-muqabala (Compendium on Calculation by Completion and Reduction).

In this title, “al-jabr”—the origin of the word “algebra”—means “restoration” or “completion” and refers to moving a negative quantity to the other side of an equation where it becomes positive. Al-muqabala means “comparison” or “reduction” and refers to the possibility of subtracting like terms from two sides of an equation. While all algebra problems were stated and solved using words and sentences—symbolic algebra did not arise until much later in the fifteenth century in Italy—an algebra of polynomials was developed by Abu Kamil (c. 850–930), Karaji (c. 953–1029), and Samu’il Maghribi (c. 1130–1180+, also known as al-Samaw’al). Powers, even negative powers, of unknowns were considered and many algebraic equations were classified and solved. Khwarizmi gave a full account of second-degree equations, and Khayyam (1048–1131) gave a geometric solution to equations of degree three using conic sections. Here, we give a problem—translated to modern notation—solved by Abu Kamil. Some 300 years later, this exact same problem appeared in Chapter 15 of the 1202 text Liber Abaci by Leonardo Fibonacci. Abu Kamil gave a solution to the following system of three equations and three unknowns:

Abu Kamil first started with the choice of x = 1 and solved the latter two equations for y and z. Since, for the latter two equations, any scalar multiple of the solutions continues to be a solution, he then scaled the solutions so that the first equation was also satisfied. He simplified the answer to get:

Geometry

Geometrical methods and problems were ubiquitous in the Islamic era. While algebraic problems were solved using the newly developed algebraic algorithms (the word “algorithm” itself is derived from algorismi, the Latin version of the name of the mathematician al-Khwarizmi), the justification for the algebraic methods was usually given using geometrical arguments and often relying on a distinctively Euclidean style. Guided by problems in astronomy and geography (for example, finding, from any place on Earth, the direction of Mecca for the purpose of the Islamic daily prayers), spherical geometry was developed.

But new work in plane geometry was also carried out. Khayyam and Nasir al-din Tusi (1201–1274), for example, studied the fifth postulate of Euclid and came close to ideas that much later on led to the development of non-Euclidean geometries in Europe. However, as is the case with much of the mathematics of this era, applications play an important role in the choice of questions and problems.

For example, Abu’l Wafa Buzjani (940–997) reports on meetings that included mathematicians and artisans. A problem of interest to tile makers is how to create a single square tile from three tiles. A traditional mathematician, Abu’l Wafa explains, translates this problem into a ruler and compass construction and gives a method for constructing a square of side √3.

While logically correct, this construction is of little use to the tile maker, who is confronted with three actual tiles and wants to cut and rearrange them to create a new tile. Abu’l Wafa also gives the customary practical method that is actually used by tile makers to solve this problem, and proves that their method, while practical, is not precise, and the final object is not exactly a square. While stressing the importance of being both practical and precise, and the virtues of Euclidean proofs, he presents his own practical and correct methods for solving this and related problems.

Trigonometry

The origins of trigonometry begin with the Greek study of chords as well as the Indian development of what is now called the “sine function.” Claudius Ptolemy’s table of chords and Indian tables of sine values were powerful tools in astronomy. However, a systematic study and use of all the trigonometric functions motivated by applications to astronomy, spherical geometry, and geography begins in the Islamic era. Abu’l Wafa had a proof of the addition theorem for sines and used all six trigonometric functions; Abu Rayhan Biruni (973–1048) used trigonometry to measure the circumference of Earth; and Nasir al-din Tusi gave a systematic treatment in his Treatise on the Quadrilateral that helped establish trigonometry as a distinct discipline.

Combinatorics

One of the earlier known descriptions and uses of the table of binomial coefficients (also known as the Pascal triangle) is that of Karaji. While his work on the subject is not extant, his clear description of the triangle survives in the writings of Samu’il Maghribi. Binomial coefficients were used extensively, among other applications, for extracting roots. Kashani, for example, used binomial coefficients to give an algorithm for extracting fifth roots. He demonstrated it by finding the fifth root of 44,240,899,506,197. Other combinatorial questions were treated as well. Ibn al-Haytham (c. 965–1039, also known as Alhazen) gave a construction of magic squares of odd order, and Ibn Mun’im (died c. 1228) devotes a whole chapter of his book Fiqh al-Hisab to combinatorial counting problems.

Numerical Mathematics

The prominence of applied problems, the development of Hindu-Arabic numerals and calculation schemes, and the development of algebra and trigonometry led to a blossoming of numerical mathematics. One prime example is Kashani’s Miftah al-Hisab or Calculators’ Key. In addition to his approximation of 2π and his extraction of fifth roots, he also gave an iterative method for finding the root of a third-degree polynomial in order to approximate the sine of one degree to as close as an approximation as one wishes.

Bibliography

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

Katz, Victor J. A History of Mathematics. An Introduction. 2nd ed. London: Addison Wesley, 1998.

Katz, Victor, ed. The Mathematics of Egypt, Mesopotamia, China, India, and Islam. A Sourcebook. Princeton, NJ: Princeton University Press, 2007.

Van Brummelen, Glen. The Mathematics of the Heavens and the Earth: The Early History of Trigonometry. Princeton, NJ: Princeton University Press, 2009.