182 CHAPTER 4 Systems of Numeration Babylonian Numerals The oldest known numeration system that resembled a place-value system was developed by the Babylonians in about 2500 b.c. Their system resembled a place-value system with a base of 60, a sexagesimal system. It was not a true place-value system because it lacked a symbol for zero. The lack of a symbol for zero led to a great deal of ambiguity and confusion. Table 4.5 gives the Babylonian numerals. The positional values in the Babylonian system are , (60) , (60) , 60, 1 3 2 … In a Babylonian numeral, a gap is left between the characters to distinguish between the various place values. From right to left, the sum of the first group of numerals is multiplied by 1. The sum of the second group is multiplied by 60. The sum of the third group is multiplied by (60) ,2 and so on. Example 1 Writing a Hindu–Arabic Numeral in Expanded Form Write 2546 in expanded form. Solution 2546 (2 10 ) (5 10 ) (4 10) (6 1) (2 1000) (5 100) (4 10) (6 1) 3 2 = × + × + × + × = × + × + × + × 7 Now try Exercise 11 Table 4.5 Babylonian Numerals Babylonian numerals Hindu–Arabic numerals 1 10 Example 2 The Babylonian System: A Place-Value System Write as a Hindu–Arabic numeral. Solution 60’s units 1010111 60’s (23 60) 1380 + 10 10 1 1 units (22 1) 22 ↓ ↓ + + + + × + + + × Thus, the answer is 1380 + 22 or 1402. 7 Now try Exercise 21 The Babylonians used the symbol to indicate subtraction. The numeral represents 10 2, − or 8. The numeral represents 40 3, − or 37 in base 10 or decimal notation. Example 3 From a Babylonian to a Hindu–Arabic Numeral Write as a Hindu–Arabic numeral. Solution The place value of these three groups of numerals from left to right is (60) , 60, 1 3600, 60, 1 2 or The numeral in the group on the right has a value of 20 2, − or 18. The numeral in the center group has a value of 10 1, + or 11. The numeral on the left represents 1 1, + or 2. Multiplying each group by its positional value gives Did You Know? Counting Boards One of the earliest counting devices, used in most ancient civilizations, was the counting board. The oldest-known counting board ever found dates to about 300 B.C. It is thought to have been used by the Babylonians for counting and for gaming. It is believed the ancient Sumerians used a counting board as early as 2500 B.C. On such a board, each column represents a positional value. The number of times a value occurs is represented by markers (beads, stones, sticks) in the column. An empty column signifies “no value.” The widespread use of counting boards throughout Europe meant that Europeans were already long accustomed to working with positional values when they were introduced to Hindu–Arabic numerals in the fifteenth century. Ekkapon/ Shutterstock
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