Islamic Civilization

Arithmetic and the Problem of Arabic Numerals

Julian A. Smith notes how until recently, the Islamic contribution to arithmetic has been generally little known, though Muslim mathematicians pioneered a number of new techniques.[1] Smith sums up some such accomplishments, including how Muslim mathematicians pioneered decimal arithmetic, and how they made considerable contributions to the ancient sexagesimal (base 60) system of arithmetic, which had been developed by the Babylonians around 2000 BC.[2] This system was widely used for astronomical calculation throughout the ancient world. Sexagesimal addition, subtraction, multiplication, and division became so commonplace among Islamic astronomers, it was renamed ‘the astronomer’s arithmetic.’[3] Muslim astronomers such as al-Kashi (1380-1429) used sexagesimal numbers to determine approximate roots, extract roots, and even find the fifth root of certain numbers. [4]

Yushkevitch, who did some pioneering works on Muslim mathematics and wrote on the particular contribution of Al-Kashi,[5] stresses the importance of Muslim accomplishments in arithmetic.[6] He too notes how in the realm of computing arithmetic, Muslims completed and perfected approximate methods for the extraction of roots, the procedures for the verification of calculations, their most favourable arrangements, and so on.[7] Al-Kashi’s  calculation of [pi] to seventeen decimal places was a brilliant example of this, and in connection with the extraction of roots, we meet for the first time in al-Kashi the binominal theorem for any positive integrer exponent.[8] The time and place of origin of this rule are unknown, but it was probably proposed by Omar  Khayyam (1048-1131).[9] The technique of operations on whole numbers and sexagesimal fractions had been perfected. Al-Kashi also introduced decimal fractions and stressed their advantages.[10]

Centuries before him, al-Karaji’s (fl. 1020) works included a manuscript on the rule of computation entitled Al-Kafi fi al-Hisab (Essential of Arithmetic).  One of the major contributions of  al-Karaji, noted by Rashed, was to conceive a new mathematical project: the arithmetisation of algebra. In the words of one commentator, he enabled the algebraist "to work with unknowns with all the arithmetic instruments, just as the arithmetician works with the knowns."[11] This involves a transposition and extension of elementary arithmetic operations—the algorithms as well as Euclidean division or the extraction of roots—to algebraic terms and expressions, and particularly to polynomials.[12] Thanks to the arithmetisation of algebra, al-Karaji arrived at the construction of the algebra of polynomials and also gained a better understanding of the algebraic structure of real numbers.[13]

Like other branches of  Islamic mathematics, Muslim arithmetic was affected by practical considerations posed by the faith such as problems of inheritance and finance, and the need to calculate events in the lunar based Islamic calendar.[14] The Islamic laws of inheritance, as found in the Qur’an,  King points out, are complicated, and their application involves skills in arithmetic, and both legal scholars and certain mathematicians wrote on this subject.[15] Al-Khwarizmi , for instance, devoted the second half of his treatise on algebra to the calculation of shares of an estate given to various heirs.[16] These problems employed the arithmetic of fractions, and were heavily influenced by religious law and customs. A typical problem treated by al-Khwarizmi involves the calculation of the shares of a dead woman’s estate that would accrue to her husband, her son, and her three daughters.[17] 

The numeral system (1, 2, 3, …) now taken for granted, remains by far one of the greatest scientific legacies to humanity. Centuries ago Arabic numerals were shunned in Western Christendom as symbols of the Muslim foe: money changers, for instance, were summoned to stick to the ancient method of ‘the fathers’ instead of making use of the Arabic numerals in their transactions.[18] Until five to six centuries ago, demand by society and science hardly required the Arabic numerals (except by traders who used them more widely), and reliance on the Roman numerals was adequate.[19] However, once we stepped into our modern times, and the decisive role of such numbers became fundamental for every calculation or operation, and they increasingly became the foundation of modern science, the Arabic/Islamic association with such numbers became less accepted. And so they began to be gradually taken away from the Muslims. These numbers, once rehabilitated, and having become the foundation of modern science, are now called Hindu Numerals , or according to a large number of Western historians, they are Greek.[20] In respect to the latter, the numerals belong to this endless list of Greek scientific achievements lost for over ten centuries until they were recovered in the 12^th-13^th century in the West. This latter Greek origin of the numbers is defended by the likes of Bubnov, who late in the nineteenth century fought against the Hindu origin of such numerals, asserting that the fundamental elements of our present system were known in classical antiquity, and that the debt is to the Greeks not to the Hindus.[21] Woepcke also argued the same origins, saying that these numerals had reached Spain through the neo Pythagoreans by c. 450 CE.[22] These once shunned numbers in Western Christendom have become, thus, products of Western culture.[23]

The situation we have today is that, these numerals, hated, despised, and rejected for centuries as a symbol of the hated Muslim foe, today are either Greek or Hindu, but definitely not ‘Arab.’ How such extremely opposed views can be reconciled is difficult to comprehend. How can the easy, obvious and simple facts that these numbers were acquired from the Muslims, that the Muslims have been using them for centuries in the western realm of Islam, and that it was the Muslims who wrote and explained their use, be rejected as weak evidence in favour of a Muslim source is equally hard to comprehend.

There is no need here to get bogged down in the argument of who invented these numerals, but for those eager to follow it, there are a number of good sources.[24] All that needs to be said here is that the numerals are not of Roman or Greek, or of European origin at all. Such evidence relies on exuberant writings. Had they been European in origin, they would have been obviously present in lands where the Romans went, or where there was Greek impact before the Muslims. Not a single piece of writing, or archaeological finding, or anything whatsoever bears any evidence of their existence in Europe before the Muslims. The numerals are not wholly Arabic either, nor are they Hindu. There is surely something Hindu (not in the form but in certain functions) and something from the Western side of the Islamic world (as these numbers were predominantly used in North Africa  and Spain). Thus, they are the outcome of a combinatory exercise of techniques and methods, which the Muslims amalgamated, developed, and adapted to their use and practical needs. They are, thus, a concoction from the Hindu and from the Muslim mathematicians themselves. As for the word Hindi related to such numerals, which has been used by some to justify the attribution to the Hindu, this has been explained by Kaye and Carra de Vaux, most particularly. For Carra de Vaux, for instance, the word Hindi is easily confused in the Arabic script with hindasi which means what relates to geometry or the art of the engineer.[25]

The most important remark to make in relation to these decimals, and here was where Islamic genius lay, is the manner by which Muslims recognised their importance earlier than anyone else. The greater Muslim contribution was in developing and shaping the use of such numerals according to uses that are not just valid in our modern world, but are its very foundations.  These numerals, furthermore, contributed decisively to the advance of mathematics and, as put by Wickens, opened a door to progress that it is difficult to imagine without them.[26] It is very difficult to see what could have been done with Roman numerals; eight (8), for example, in Roman is VIII; Forty-seven (47) is XXXXVII. One can imagine the struggle in making the simplest of calculations with such Roman numerals; complex calculations are impossible with them. In fact, the use of Roman numerals in the West, according to Watt, retarded the advance of mathematical theory.[27] The nine decimals, in addition to the Zero, in the end, in agreement with Wickens could be said to be ‘nearly as great a revolution as the art of writing.'[28]

The manner the knowledge of such decimals was transferred to us is crucial. Al-Khwarizmi ’s main work other than his algebra was a treatise on arithmetic, in which the numerals were presented and the place value system was explained. This textbook was the earliest written on the decimal system, representing a milestone in the development of mathematics and science.[29] In it, Al-Khwarizmi demonstrates the basic operations of addition, subtraction, division and multiplication, and shows how to work with fractions and how to extract square roots, all operations greatly simplified by the new system.[30] The Latin  translation of this work begins with the words Dixit algorismi, or ‘Algorithm (Al-Khwarizmi) says, and follows with instructions for making various computations, thus Algorithm, a Latinised version of Al-Khwarizmi’s name, has come to its present meaning of a general computational procedure.[31]

With regard to the transmission of the numerals to the Christian West, Sarton explains that the writings of al-Kindi and al-Khwarizmi were in all probability the main channels through which they became known in Islam and later in the West.[32] The earliest Muslim documents bearing such numerals date from 874 and 888; and their propagation must have been speeded by an exceedingly active trade, that reached every part of the world.[33] Al-Khwarizmi , according to Arndt, was enormously popular and avidly studied in the West, and was instrumental in effecting Europe’s conversion from the cumbersome Roman numerals to the present day system.[34] The first translation of Al-Khwarizmi into Latin  was by Adelard of Bath in 1126.[35] However, the earliest and best known practical attempt to introduce north of the Pyrenees Islamic mathematics, including the numeral system, was made by Gerbert of Aurillac (Pope Silvester 999-1003). This was met with great opposition, though; Gerbert’s mathematics were called ‘A Dangerous Saracen magic.'[36] By the 13^th century, things evolved, and a better appreciation of the Muslim system took place, witness one Fibonacci, who was part of the Pisan trading colony in the Algerian city of Bejaia , and who concluded the superiority of Arabic numerals and Muslim methods of calculations, and so sent his son Leonardo to learn mathematics there.[37] The outcome of such studies was The Liber Abacci of Leonardo of Pisa, whose second edition would establish the ‘Andalusian number system as the basis of modern mathematics.’[38] It seems, though, that a much more popular work than Leonardo’s that employs the Arabic numerals was the Algorismus of John of Holywood, who was a teacher in Paris.[39] John of Holywood’s (d. 1250) book on arithmetic, or rather ‘algorisms,’ was extremely popular, and did more to introduce the Muslim notation than any other, and was frequently re-printed down to the 17^th century.[40] Wright, however, notes the slow progress of the use of the numerals,[41] and Sir Hilary Jenkinson reminds us that until well into the 17^th century the counting board and the old method of digital computation were in general use in England, and that until the very end of this century writing masters did not include copies of Arabic figures in their books.[42] After that they increasingly became the foundation of all calculations, modern science and civilisation, and, of course, simultaneously, also began to lose their ‘Arabic’ parenthood.