4.1 Additive, Multiplicative, and Ciphered Systems of Numeration 175 Multiplicative Systems Multiplicative numeration systems are more similar to our Hindu–Arabic system than are additive systems. In a multiplicative system, 642 might be written (6)(100) (4)(10)(2) or 6 100 4 10 2 Note that no addition signs are needed in the representation. From this illustration, try to formulate a rule explaining how multiplicative systems work. The principal example of a multiplicative system is the traditional Chinese system. The numerals used in this system are given in Table 4.3. The numeration system used in China today is different from the traditional system discussed in this chapter. The present-day numeration system in China is a positional-value system rather than a multiplicative system, and in some areas of China, 0 is used as the numeral for zero. In the Roman numeration system, a symbol is not repeated more than three consecutive times. For example, the number 646 would be written DCXLVI instead of DCXXXXVI. The second advantage of the Roman numeration system over the Egyptian numeration system is that it makes use of the multiplication principle for numerals greater than 1000. A bar above a symbol or group of symbols indicates that the symbol or symbols are to be multiplied by 1000. Thus, V 5 1000 5000, = × = X 10 1000 10,000, = × = and CD 400 1000 400,000. = × = Other examples are VI 6 1000 6000, = × = XIX 19 1000 19,000, = × = and XCIV 94 1000 94,000. = × = This practice greatly reduces the number of symbols needed to write large numbers. Example 5 From a Hindu–Arabic Numeral to a Roman Numeral Write 439 as a Roman numeral. Solution 439 400 30 9 = + + To represent 400, we will subtract 100 (C) from 500 (D). We will place C to the left of D to indicate the subtraction (500 100). − Likewise, to represent 9, we will subtract l (I) from 10 (X). We will place I to the left of X to indicate the subtraction (10 1). − To represent 30 we will simply write X three times. Therefore, 439 400 30 9 (500 100) 10 10 10 (10 1) = ++= − ++++ − CDXXXIX = 7 Now try Exercise 33 Did You Know? An Important Discovery Archaeologist Denise Schmandt- Besserat made a breakthrough discovery about early systems of numeration. She realized that the little clay geometric objects that had been found in many archaeological sites had actually been used by people to account for their goods. Later in history, these tokens were impressed on a clay tablet to represent quantities, the beginning of writing. Example 6 Writing a Large Roman Numeral Write 12,345 as a Roman numeral. Solution 12,345 (12 1000) 300 40 5 = × + + + (10 1 1) 1000 (100 100 100) (50 10) 5 [ ] = + + × + + + + − + XIICCCXLV = 7 Now try Exercise 43 Write a Hindu–Arabic Numeral as a Roman Numeral
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