192 CHAPTER 4 Systems of Numeration the duodecimal system , we use the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, and B, where A represents ten and B represents eleven. For base 16, known as the hexadecimal system, we use the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. Example 7 Converting to and from Base 16 a) Convert 7DE16 to base 10. b) Convert 6713 to base 16. Solution a) In a base 16 system, the positional values are , 16 , 16 , 3 2 … 16, 1 or , 4096, … 256, 16, 1. Since D has a value of thirteen and E has a value of fourteen, we perform the following calculation. 7DE (716)(D16) (E1) (7 256) (13 16) (14 1) 1792 208 14 2014 16 2 = × + × + × = × + × + × = + + = b) The highest power of base 16 less than or equal to 6713 is 16 ,3 or 4096. When we divide, if we obtain a quotient greater than nine but less than sixteen, we will use the corresponding letter A through F. 6713 4096 1 with remainder 2617 2617 256 A with remainder 57 57 16 3 with remainder 9 ÷ = ÷ = ÷ = Note that A has a value of ten. Thus, 6713 1A39 . 16 = 7 Now try Exercise 35 Timely Tip It is important to remember the following items presented in this section. ■ If a numeral is shown without a base, we assume the numeral is a base 10 numeral. ■ When converting a base 10 numeral to a different base, your answer should never contain a digit greater than or equal to that different base. ■ When changing a numeral given in a base other than 10 to a numeral in base 10, use multiplication. ■ When changing a base 10 numeral to a numeral in a different base, use division. Example 6 Converting to and from Base 12 a) Convert 39BA12 to base 10. b) Convert 6893 to base 12. Solution a) In base 12, the positional values are , 12 , 12 , 12, 1 3 2 … or , 1728, 144, 12, 1. … Since B has the value of eleven and A has the value of ten, we perform the following calculation. 39BA (3 12 ) (9 12 ) (B 12) (A 1) (31728) (9144)(1112)(101) 5184 1296 132 10 6622 12 3 2 = × + × + × + × = × + × + × + × = + + + = b) The highest power of base 12 that is less than or equal to 6893 is 12 ,3 or 1728. To convert 6893 to base 12, we will use a process similar to the one we used in Examples 4 and 5. Remember that in base 12 we represent ten with the numeral A and eleven with the numeral B. ÷ = ÷ = ÷ = 6893 1728 3 with remainder 1709 1709 144 B with remainder 125 125 12 Awith remainder 5 Note that B has a value of eleven. Note that A has a value of ten. The remainder, 5, is less than the base, 12, so no further division is necessary. Thus, the answer is 3BA5 . 12 To check this answer, perform the calculation (3 12 ) (11 12 ) (10 12) (5 1) 3 2 × + × + × + × to verify that you obtain 6893 in base 10. 7 Now try Exercise 33
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