4.3 Other Bases 191 Therefore, there are seven groups of 82 in 486. Next divide the remainder, 38, by 8. Second digit in answer 38 8 4 with remainder 6 ÷ = Third digit in answer There are four groups of 8 in 38 and 6 units remaining. Since the remainder, 6, is less than the base, 8, no further division is required. (7 64) (4 8) (6 1) (7 8) (4 8) (6 1) 746 2 8 = × + × + × = × + × + × = Notice that we placed the subscript 8 to the right of 746 to show that it is a base 8 numeral. 7 Now try Exercise 23 The same procedure is used in Example 5, but it is given in a simplified manner. Example 5 Converting from Base 10 to Base 3 Convert 273 to base 3. Solution The place values in the base 3 system are , 3 , 3 , 3 , 3 , 3 , 3, 1, 6 5 4 3 2 … or , 729, 243, 81, 27, 9, 3, 1. … The highest power of the base that is less than or equal to 273 is 3 ,5 or 243. Successive divisions by the powers of the base give the following result. 273 243 1 with remainder 30 30 81 0 with remainder 30 30 27 1 with remainder 3 3 9 0 with remainder 3 3 3 1 with remainder 0 ÷ = ÷ = ÷ = ÷ = ÷ = The remainder, 0, is less than the base, 3, so no further division is necessary. To obtain the answer, list the quotients from top to bottom followed by the remainder in the last division. We can represent 273 as one group of 243, no groups of 81, one group of 27, no groups of 9, one group of 3, and no units. =× +× +× +× +×+× =× +× +× +× +×+× = 273 (1 243) (0 81) (1 27) (0 9) (1 3) (0 1) (1 3 ) (0 3) (1 3) (0 3) (1 3) (0 1) 101010 5 4 3 2 3 7 Foreverhappy/123RF Now try Exercise 25 Bases Greater Than 10 Recall that a place-value system with base b must have symbols for the digits from 0 up to one less than the base. For example, base 6 uses the digits 0, 1, 2, 3, 4, and 5; and base 8 uses the digits 0, 1, 2, 3, 4, 5, 6, and 7. What happens if the base is larger than ten? We will need single-digit symbols to represent the numbers ten, eleven, twelve, . . . up to one less than the base. In this textbook, whenever a base larger than ten is used we will use the capital letter A to represent ten, the capital letter B to represent eleven, the capital letter C to represent twelve, and so on . For example, for base 12, known as Did You Know? Dozenal Society The Dozenal Society of America is a voluntary nonprofit organization that conducts research and educates the public in the use of the base 12 numeration system. Throughout history, 12 has been used for many groupings of items. Some familiar examples are 12 inches in a foot, 12 hours on a clock, 12 months in a year, 12 items in a dozen, and 12 dozen in a gross. The Dozenal Society’s mascot is the panda “because it has six ‘fingers’ on each paw, and if we had been so blessed we would be counting in the more convenient dozenal base today.” For more information on the Dozenal Society, you can visit their website at Dozenal.org.
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