174 CHAPTER 4 Systems of Numeration Users of additive systems easily accomplished addition and subtraction by combining or removing symbols. Multiplication and division were more difficult; they were performed by a process called duplation and mediation (see Section 4.5). The Egyptians had no symbol for zero, but they did have an understanding of fractions. The symbol was used to take the reciprocal of a number; thus, meant 1 3 and was . 1 11 Writing large numbers in the Egyptian hieroglyphic system takes longer than in other systems because so many symbols have to be listed. For example, 45 symbols are needed to represent 99,999. The Roman numeration system, a second example of an additive system, was developed later than the Egyptian system. Roman numerals (Table 4.2) were used in most European countries until the eighteenth century. They are still commonly seen on buildings, on clocks, and in books. Roman numerals are selected letters of the Roman alphabet. Roman numerals I V X L C D M Hindu–Arabic numerals 1 5 10 50 100 500 1000 Table 4.2 Roman Numerals Timely Tip When working with Roman numerals, we work from left to right . We add each numeral unless its value is smaller than the value of the numeral to its right. In that case, we subtract its value from the value of the numeral to its right. The Roman system has two advantages over the Egyptian system. The first is that it uses the subtraction principle as well as the addition principle. Starting from the left, we add each numeral unless its value is smaller than the value of the numeral to its right. In that case, we subtract its value from the value of the numeral to its right. Only the numbers 1, 10, 100, 1000, … can be subtracted, and they can only be subtracted from the next two higher numbers. For example, C (100) can be subtracted only from D (500) or M (1000). The symbol DC represents 500 100, + or 600, and CD represents 500 100, − or 400. Similarly, MC represents 1000 100, + or 1100, and CM represents 1000 100, − or 900. In addition, CX represents 100 l0, or 110, + and XC represents 100 10, or 90. − Also, XI represents 10 1, or 11, + and IX represents 10 1, or 9. − Learning Catalytics Keyword: Angel-SOM-4.1 (See Preface for additional details.) Example 3 From a Roman Numeral to a Hindu–Arabic Numeral Write MDCCLXXXVI as a Hindu–Arabic numeral. Solution Since each numeral is larger than the one to its right, no subtraction is necessary. MDCCLXXXVI 1000 500 100 100 50 10 10 10 5 1 1786 = + + + ++++++ = 7 Now try Exercise 21 Write a Roman numeral as a Hindu–Arabic numeral Example 4 From a Roman Numeral to a Hindu–Arabic Numeral Write CMLXIV as a Hindu–Arabic numeral. Solution As you read the numerals from left to right, you will note that C (100) has a smaller value than M (1000) and that C is to the left of M. Therefore, CM represents 1000 100, − or 900. Also note that I (1) has a smaller value than V (5) and that I is to the left of V. Therefore, IV represents 5 1, − or 4. The rest of the numerals can be added from left to right. CMLXIV (1000 100) 50 10 (5 1) 964 = − + + + − = 7 Now try Exercise 23
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