Lecture 19. Hindu-Arabic Numeral System The Roman numerals Roman numerals originated in ancient Rome. The Roman numeral system is a cousin of the Etruscan numerals, and the letters derive from earlier non-alphabetical symbols. The system was modified slightly during the Middle Ages to produce the system used today. The first ten Roman numerals are I;II;III;IV;V;VI;VII;VIII;IX;X: Figure 19.1 Roman numerals In general, Roman numerals are written as combinations of the seven letters in the table below. The letters can be written as capital or lower-case letters. I = 1 C = 100 V = 5 D = 500 X = 10 M = 1000 L = 50 119 There is no zero in Roman numerals. For examples, XX = 20, CC = 200, DC = 500 + 100 = 600 If smaller numbers follow larger numbers, the numbers are added. If a smaller number precedes a larger number, the smaller number is subtracted from the larger. For example, if you want to say 1; 100 in Roman Numerals, you would say M for 1000 and then put a C after it for 100; in other words 1,100=MC in Roman Numerals. For examples: ∙ VIII = 5 + 3 = 8 ∙ IX = 10 − 1 = 9 ∙ XL = 50 − 10 = 40 ∙ XC = 100 − 10 = 90 ∙ MCMLXXXIV = 1000 + (1000 − 100) + 50 + 30 + (5 − 1) = 1984 How complicated to use Roman numerals ! Roman numerals are complicated. For examples, ∙ 327 = CCCXXVII ∙ 3888 = MMMDCCCLXXXV III If one wants to compute 4 × 235 =? ∙ Step 1. Use 4 times 200 to get 800, namely, IV multiplies CC to get CCCCCCCC. ∙ Step 2. Simplify the above result to obtain DCCC. ∙ Step 3. Use 4 times 30 to get 120, namely, IV multiplies XXX to get XXXXXXXXXXXX. ∙ Step 4. Simplify the above result to obtain CXX. ∙ Step 5. Add 800 and 120 to get 920, namely, add DCCC to CXX to obtain DCCCCXX. ∙ Step 6. Add 4×5 = 20 to obtain the final answer 920, namely, add XX to DCCCCXX to obtain DCCCCXXXX. 120 Figure 19.2 Medieval education Tobias Dantzig 1 once told a story: \There is a story of a German merchant of the fifteenth century, which I have not succeeded in authenticating, but it is so characteristic of the situation then existing that I cannot resist the temptation of telling it. It appears that the merchant had a son whom he desired to give an advanced commercial education. He appealed to a prominent professor of a university for advice as to where he should send his son. The reply was that if the mathematical curriculum of the young was to be confined to adding and subtracting, he perhaps could obtain the instruction in a German university; but the art of multiplying and dividing, he continued, had been greatly developed in Italy, which in his opinion was the only country where such advanced instruction could be obtained. " The origin of the Hindu - Arabic numerals The Hindu-Arabic numeral system is a decimal place-value numeral system. It requires a zero to handle the empty powers of ten (as in \205"). With the nine figures 1, 2, ..., 9 and the symbol 0, any number can be represented easily. This is the numeral system that we are using today. The origin of this decimal place value system is supposed in India and its transmission to the West via the Arabs. However, the actual origins of the important components of this system, the digits 1 through 9 themselves, the notion of place value, and the use of 0, are to some extend lost to the historical record. 2 1Tobias Dantzig, Number: The Language of Science, Plume, a member of Penguin Group(USA), 2007, p. 26. 2Victor J. Katz, A History of Mathematics - an introduction, 3rd editions, Addison -Wesley, 2009; p.233. 121 ∙ The Babylonians had a place value system; the Chinese had a multiplicative system with base 10; in India, there were number symbols to represent 1 through 9 and also to represent 10 through 90. ∙ Around the the year 600, the Indians evidently dropped the symbols for numbers higher than 9 and began to use their symbols for 1 through 9. In a fragment of a work of Severus Sebokht, a Syrian priest, dated 662, is the remark that the Hindus have a valued method of calculation \done by means of nine signs," but he did not mention a sign of zero. 3 ∙ Evidence of early use of a zero glyph may be present in Bakhshali manuscript, a text of poor condition discovered in 1881. The best evidence we have is that this manuscript dates from the seventh century. The \zero" was denoted as a dot there. There is a possibility that Severus did not consider the dot as a \ sign." ∙ The earliest dated inscriptions using the decimal place value system including the zero were found in Cambodia, dated 683. The dot as system for 0 as part of a decimal place value system also appeared in Chiu-Chih Li, the Chinese astronomical work of 718. Figure 19.3 Hindu-Arabic numbers Some people thought that the decimal system is very ancient, and so is the position system; but their combination appears in China and then in India 4. In any case, it is sure that the decimal place value system was fully developed in India by the 8th century. 3Victor J. Katz, A History of Mathematics - an introduction, 3rd editions, Addison -Wesley, 2009; p.233. 4D. J. Struik, A Concise History of Mathematics, the 4th edition, Dover Publications, Inc., 1987, p. 67; or A History of Mathematics - an introduction, 3rd editions, Addison -Wesley, 2009; p.235. 122 It was in the 12th century, the Arabic numeral system was introduced to the western world through Latin translations. Spreading Hindu - Arabic numeral system in Europe Leonardo Pisano Bogollo, (1170-1250) also known as Leonardo of Pisa, or, simply Fibonacci, was an Italian mathe- matician. Fibonacci is best known to the modern world for the spreading of the Hindu-Arabic numeral system in Europe, primarily through the publication in the early 13th century of his Book of Calculation, the Liber Abaci. 5 Leonardo introduced the Hinda - Arabic numerals to the west. He wrote in his book at the beginning: \There are nine figures of the Indian 1 2 3 4 5 6 7 8 9. With these nine figures and the symbol 0, which in Arabic is called zephirum, any number can be written as ......"6 This is the first time a European mathematician described zero. Comparing the calculation of Roman numerals last section, it is obvious that the Hinda - Arabic numerals has much more advantages than the Roman one. Nevertheless, more than 700 years ago, people did not think so. For many years, account books were still kept in Roman numerals. It was believed that the Hindu-Arabic numerals could be altered too easily, and thus it was risky to depend on them alone in recording large commercial transaction. 7 In 1298, the city council of Florence, Italy, banned the use of zero entirely. 8 Sometimes during the 14th century Italian merchants began to use some Arabic figures in their account books. 9 In La disme (1585), Simon Steven introduced decimal fractions as part of project to unify the whole system of measurements on a decimal base. It was one of the great improvement made possible by the general introduction of the Hindu-Arabic system of numeration. 5A number sequence named after him known as the Fibonacci numbers, which he did not discover but used as an example in the Liber Abaci. 6Art Johnson, Famous problems and their mathematics, Greenwood publisher group, 1999, p.44. 7Victor J. Katz, A History of Mathematics - an introduction, 3rd editions, Addison -Wesley, 2009, p.385. 8Art Johnson, Famous problems and their mathematics, Greenwood publisher group, 1999, p.44. 9D. J. Struik, A Concise History of Mathematics, the 4th edition, Dover Publications, Inc., 1987, p.81. 123.