4.1 POSITIONAL NUMBER SYSTEMS 181 (Web design course) Our decimal notation,
4.1 POSITIONAL NUMBER SYSTEMS 181 Our decimal notation, which differs from the more ancient forms primarily because of its fixed radix point, together with its symbol for zero to mark an empty position, was developed first in India within the Hindu culture. The exact date when this notation first appeared is quite uncertain; about 600 A.D. seems to be a good guess. Hindu science was rather highly developed at that time, particularly in astronomy. The earliest known Hindu manuscripts that show this notation have numbers written backwards (with the most significant digit at the right), but soon it became standard to put the most significant digit at the left. About 750 A.D., the Hindu principles of decimal arithmetic were brought to Persia, as several important works were translated into Arabic; a picturesque account of this development is given in a Hebrew document, which has been translated into English in AA4M 15 (1918), 99-108. Not long after this, al- Khwarizmi wrote his Arabic textbook on the subject. (As noted in Chapter 1, our word algorithm comes from al-Khwbrizmi s name.) His work was trans- lated into Latin and was a strong influence on Leonardo Pisano (Fibonacci), whose book on arithmetic (1202 A.D.) played a major role in the spreading of Hindu-Arabic numerals into Europe. It is interesting to note that the left-to-right order of writing numbers was unchanged during these two transitions, although Arabic is written from right to left while Hindu and Latin scholars generally wrote from left to right. A detailed account of the subsequent propagation of decimal numeration and arithmetic into all parts of Europe during the period from 1200 to 1600 A.D. has been given by David Eugene Smith in his History of Mathematics 1 (Boston: Ginn and Co., 1923), Chapters 6 and 8. Decimal notation was applied at first only to integer numbers, not to frac- tions. Arabic astronomers, who required fractions in their star charts and other tables, continued to use the notation of Ptolemy (the famous Greek astronomer), a notation based on sexagesimal fractions. This system still survives today, in our trigonometric units of degrees, minutes, and seconds, and also in our units of time, as a remnant of the original Babylonian sexagesimal notation. Early European mathematicians also used sexagesimal fractions when dealing with noninteger numbers; for example, Fibonacci gave the value 1 22 7 42 33N 4v 40 as an approximation to the root of the equation x3 + 2s2 + lOa: = 20. (The correct answer is lo 22 7 42 33N 4 38M 30W1 50* 151x 43x . . . .) The use of decimal notation also for tenths, hundredths, etc., in a similar way seems to be a comparatively minor change; but, of course, it is hard to break with tradition, and sexagesimal fractions have an advantage over decimal fractions in that numbers such as 4 can be expressed exactly, in a simple way. Chinese mathematicians-who never used sexagesimals-were apparently the first people to work with the equivalent of decimal fractions, although their numeral system (lacking zero) was not originally a positional number system in the strict sense. Chinese units of weights and measures were decimal, so that Tsu Chhung-Chih (who died c. 500 A.D.) was able to express an approximation