4.1 Additive, Multiplicative, and Ciphered Systems of Numeration 173 Four types of numeration systems used by different cultures are the topic of this chapter. They are additive (or repetitive), multiplicative, ciphered, and place-value systems. You do not need to memorize all the symbols, but you should understand the principles behind each system. By the end of this chapter, we hope that you better understand the system we use, the Hindu–Arabic system , and its relationship to other types of systems. Additive Systems An additive system is one in which the number represented by a particular set of numerals is simply the sum of the values of the numerals. The additive system of numeration is one of the oldest and most primitive types of numeration systems. One of the first additive systems, the Egyptian hieroglyphic system, dates back to about 3000 b.c. The Egyptians used symbols for the powers of 10: 100 or 1, 101 or 10, 102 or 10 10, 103 × or 10 10 10, × × and so on. Table 4.1 lists the Egyptian hieroglyphic numerals with the equivalent Hindu–Arabic numerals. To write the number 600 in Egyptian hieroglyphics, we write the numeral for 100 six times: . Table 4.1 Egyptian Hieroglyphics Hindu–Arabic Numerals Egyptian Numerals Description 1 Staff (vertical stroke) 10 Heel bone (arch) 100 Scroll (coiled rope) 1000 Lotus flower 10,000 Pointing finger 100,000 Tadpole (or whale) 1,000,000 Astonished person Did You Know? The Rhind Papyrus Album/Alamy Stock Photo Much of our knowledge of Egyptian mathematics and numeration comes from a roll of papyrus measuring 18 ft by 13 in. Written around 1650 B.C. , it was discovered in a shop in Luxor, Egypt, in 1858 by Henry Rhind, a Scottish lawyer turned archaeologist. Unlike the straightforward accounting of property and events common to Egyptian tombs, the Rhind Papyrus has inscribed on it 85 mathematical problems and solutions involving addition, subtraction, multiplication, division, and geometry. The key to translation of the papyrus was the Rosetta Stone, which had been discovered some 60 years earlier by one of Napoleon’s officers. Both of these treasures now reside at the British Museum in London. Example 1 From an Egyptian to a Hindu–Arabic Numeral Write the following numeral as a Hindu–Arabic numeral. Solution 10,000 10,000 100 100 100 10 1 1 1 1 + + + + + + + + + 20,314 = 7 Now try Exercise 9 Example 2 From a Hindu–Arabic to an Egyptian Numeral Write 1,203,462 as an Egyptian numeral. Solution 1,203,462 1,000,000 200,000 3000 400 60 2 = + + + + + 7 Now try Exercise 15 In the Egyptian hieroglyphic system, the order of the symbols is not important. For example, and both represent 100,212.
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