The Use of Hindu-Arabic Numerals Aids Mathematicians and Stimulates Commerce

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The Use of Hindu-Arabic Numerals Aids Mathematicians and Stimulates Commerce

Overview

The modern system of notation, using ten different numerals including a zero and using position to denote value, appears to be the invention of Hindu mathematicians and astronomers, reaching its present form by the seventh century. The system became known in western Europe through the works of Islamic commentators whose works were translated into Latin. The Hindu-Arabic numerals, as they are now known, greatly facilitated arithmetic computations, particularly multiplication and division. They also allowed more rapid calculation of the mathematical tables needed for surveying, navigation, and the keeping of financial records and thus contributed to the extensive exploration and the growth of capitalism that characterized the Renaissance.

Background

One of the essential requirements for any type of mathematics is a means of representing quantities. At first, tokens might be used—pebbles or small clay objects with one pebble representing each sheep in a herd, for example. Eventually, tokens of different shapes might be used to represent certain multiples, say five or ten sheep instead of one. With the invention of writing, it made more sense to use marks pressed into clay or made on paper or papyrus to keep track of one's possessions. The Babylonians, Egyptians, Indians, and Mayans all had developed elaborate systems to represent quantities by the first century a.d. The Babylonians developed a sophisticated number system based on the number 60, using it in commerce and for astronomy and astrology. By the last century b.c., this system included a symbol for zero, which was used as a placeholder in expressing quantities.

Of the several number systems, those that had the greatest effect on the development of mathematics in Europe of the Middle Ages were the Roman, the Chinese, and the Indian or Hindu, transmitted to the Western world by the Arabs and now known as Hindu-Arabic numerals. In the Roman system, still occasionally used today, letters of the alphabet were used to represent units and multiples of five or ten. In the Roman system the year 2004 can be written quite compactly as MMIV, with a hint of positional notation in that the "I" appearing before the "V" means that the one it represents is to be subtracted from the five represented by the "V." Roman numerals were adequate for record keeping and could be added and subtracted easily, but were far more cumbersome in multiplication and division and certainly not suited to the needs of modern science or commerce.

The Chinese system was not a decimal system, based on the number 10, but a centesimal system, based on separate symbols for the whole numbers between 1 and 9 and for multiples of 10 between 10 and 90. By alternating the pairs of symbols, the Chinese were able to represent numbers of any size. Because the Chinese number symbols were composed of single strokes, it was possible to represent them by short sticks, and to do arithmetic by moving sticks about according to preset rules. This led in the Middle Ages to the use of counting boards by merchants to do simple arithmetic. The Chinese were also responsible for the abacus, an arrangement of beads on wires that facilitated the ordinary operations of addition, subtraction, and multiplication.

The number system we use today, based on the numerals 0-9 and using position to denote different powers of 10, originated in India. Mathematical thinking in India dates back to at least 800 b.c. Number symbols first appear in the third century b.c., including among many alternatives the so-called Brahmi symbols, which include separate symbols for the numerals 1 through 9 and the multiples of 10 from 10 to 90. The Brahmi figures gradually evolved into the "1,2,3..." of today. By the year 600, they had come to predominate and to include a symbol for zero and for the use of positional notation. The Brahamasphuta Siddhanta, a treatise on astronomy written by the astronomer and mathematician Brahmagupta (598-c. 665) includes a treatment of arithmetic using the system of ten numerals, including zero, along with rules for fractions, the computation of interest, and rules for using negative numbers. Interestingly, Brahmagupta appeared to treat the fraction 0/0 as equal to zero, and avoided the question of dividing other numbers by zero.

Beginning in a.d. 632 Arab armies expanding from the Arabian peninsula established an Islamic empire that would stretch as far eastward as India and as far west as Spain. In 755 it split into two kingdoms, one with its capital at Baghdad. There, Hindu scientists and mathematicians found themselves welcome, despite their different religious beliefs. In Baghdad they could meet the descendants of the Greek scholars who had fled to Persia, bringing their mathematical interests, after the Emperor Justinian closed Plato's academy in a.d. 529. By 766 some Hindu mathematical work had been translated into Arabic. At Baghdad the Caliph al-Ma'mun (786-833) established a "House of Wisdom" modeled on the earlier Greek academy at Alexandria, with a library and observatory.

One of the scholars at the House of Wisdom was al-Khwarizmi (c. 780-c. 850). Al-Khwarizmi's book on algebra, popularly known as the al-jabr, was translated into Latin in 1145 by Robert of Chester (fl. c. 1141-1150), an English scholar living in Islamic Spain. Al-Khwarizmi would also become known to the Western world for a book known only in Latin translation as the Al-goritmi de numero Indorum, or Al-Khwarizmi on the Hindu Method of Calculation, in which he explains the Hindu number system and how it can be used in arithmetic calculations. It is from the title of this book that we obtain the word "algorithm" for any systematic method of calculation.

Among the readers of this book were Leonardo of Pisa (c. 1170- c. 1250), also known as Leonardo Fibonacci, an Italian who traveled throughout northern Africa and became familiar with the Arab system of numbers and methods of calculation. In 1202 he wrote the Liber Abaci or Book of Calculations, in which he described the Arabic system of numbers. Although the Hindu-Arabic system of numbers was not entirely unknown in Europe, it was Fibonacci's book that led to its widespread adoption in commerce and record keeping.

Impact

The most powerful aspect of the Hindu-Arabic system is the existence of a separate numeral for zero that can serve both as a placeholder and as a symbol for "none." The zero also appears, apparently independently, in the Chinese number system and in the system developed by the Mayans of Central America.

The use of Hindu-Arabic numerals and positional notation has become so prevalent in the modern world that it is difficult to imagine what mathematics would be like without it. On the practical side, banking, accounting, and capitalism in general could hardly be possible without a system that allowed the expression of large numbers in compact form and the easy calculation of interest. While merchants could perform the required calculations for a purchase or sale using the abacus or a counting board, the new method was faster and left a permanent record. The sort of record keeping required by bankers who accepted funds on deposit, and then lent or invested the money, required a lasting and compact notation, which the Hindu-Arabic system provided.

The more efficient system of arithmetic greatly facilitated the calculation of mathematical and astronomical tables that were used in surveying, construction, navigation, and the casting of astrological horoscopes. Prior to the twentieth century and the development of the electronic computer, all such tables had to be calculated by hand by specially trained human "computers."

The decimal system, based on ten distinct digits, is of course a reflection of human anatomy with its ten fingers. One disadvantage to the system is that the number ten is divisible only by two and five. From time to time individuals have advocated a duodecimal system based on the number twelve, which is divisible by two, three, four, and six. King Charles XII of Sweden, who reigned from 1697 to 1718, even contemplated introducing such a system in his kingdom but did not proceed. For electronic computer applications, in which numbers are stored and represented by "two-state" devices, each a sort of switch that can be on or off, the binary system in which numbers are represented by ones and zeros is ideal. While this system uses only two digits, the principle of number construction and the algorithms for the arithmetic operations parallel those for the decimal system based on Hindu-Arabic numerals.

DONALD R. FRANCESCHETTI