4.1 Additive, Multiplicative, and Ciphered Systems of Numeration 177 Ciphered Systems A ciphered numeration system is one in which there are numerals for numbers up to and including the base and for multiples of the base. The numbers represented by a particular set of numerals is the sum of the values of the numerals. Ciphered numeration systems require the memorization of many different symbols but have the advantage that numbers can be written in a compact form. The ciphered numeration system that we discuss is the Ionic Greek system (Table 4.4). The Ionic Greek system was developed in about 3000 b.c., and it used letters of the Greek alphabet for numerals. Other ciphered systems include the Hebrew, Coptic, Hindu, Brahmin, Syrian, Egyptian Hieratic, and early Arabic systems. The classic Greek alphabet contains only 24 letters; however, 27 symbols were needed. Thus, the Greeks used three obsolete Greek letters— , , and —that are not part of the classic Greek alphabet. To distinguish words from numerals, the Greeks would place a mark similar to our apostrophe to the right and above each letter that was used as a numeral. In this text, we will not use this mark because we will not be using Greek words and numerals together. Example 8 Traditional Chinese Numerals Write the following as traditional Chinese numerals. a) 7080 Now try Exercise 57 b) 7008 Solution In part (a), there is one zero between the 7 and the 8. In part (b), there are two zeros between the 7 and the 8. As mentioned previously, the symbol for zero is used only once in each of these numerals. a) 7080 = 7 1000 } × } 0 100 × 8 10 } × 7 1000 } × b) 7008 = 0 100, × 0 10 × } } 8 7 Timely Tip Notice the difference between our Hindu–Arabic numeration system, which is a positional numeration system, and the Chinese system, which is a multiplicative numeration system. Below we show how to write 5678 as a Chinese numeral if the Chinese system were a positional value system similar to ours. Multiplicative Positional Value 5 5 1000 6 6 7 100 8 7 10 8 Note that the multiples of base 10 are removed when writing positional value numerals. We will discuss positional value systems in more detail shortly.
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