194 CHAPTER 4 Systems of Numeration In Exercises 53–56, suppose colors as indicated below represent numerals in a base 4 numeration system. 0 = 1 = 2 = 3 = Write each of the following numerals as a base 10 HinduArabic numeral. 53. 4 7 54. 4 14 55. 4 36 56. 4 57 In Exercises 57– 60, write the Hindu–Arabic numerals in the base 4 numeration system discussed in Exercises 53–56. You will need to use the colors indicated above to write the answer. 57. 10 * 58. 15 * 59. 60 * 60. 56 * 61. Another Conversion Method There is an alternative method for changing a base 10 numeral to a different base. This method will be used to convert 328 to base 5. Dividing 328 by 5 gives a quotient of 65 and a remainder of 3. Write the quotient below the dividend and the remainder on the right, as shown. 5328 65 remainder 3 Continue this process of division by 5. 5328 remainder 5 65 5 13 5 2 0 64. a) What is 112 equal to in base 10? 3 b) What is 118 equal to in base 10? 9 c) What is 1116 equal to in base 10? 17 d) What is 1132 equal to in base 10? 33 e) In general, for any base b, what is 11b equal to in base 10? b 1 + Challenge Problems/Group Activities 65. Determine b if 111 43. b = b 6 = 66. Determine d if ddd 124. 5 = d 4 = 67. a) Use the numerals 0, 1, and 2 to write the first 20 numbers in the base 3 numeration system. * b) What is the next numeral after 222 ?3 10003 68. Computer Code The ASCII code used by most computers uses the last seven positions of an eight-bit byte to represent all the characters on a standard keyboard. How many different orderings of 0’s and 1’s (or how many different characters) can be made by using the last seven positions of an eight-bit byte? 2 128 7 = 69. a) Your Own Numeration System Make up your own base 20 positional-value numeration system. Indicate the 20 numerals you will use to represent the 20 numbers less than the base. Answers will vary. b) Write 523 and 5293 in your base 20 numeration system. Answers will vary. Recreational Mathematics 70. The Price Is Right Refer to the Recreational Math box on page 190. Determine the correct order in which to place the digits 0, 5, 5, 9 to match the price of the 2023 Sunfish Laser Performance sailboat. m Place the digits 0 5 5 9 , , , in the correct order to match the price of the sailboat. $ ___ ___ ___ ___ $ 5950 71. Suppose a base 4 place-value system has its digits represented by colors as follows: 0 = 1 = 2 = 3 = a) Determine the value of 4 in base 10. 876 b) Write 177 in the base 4 system using only the four colors given in the exercise. * 3 0 3 2 Answer (In the last division, since the dividend, 2, is smaller than the divisor, 5, the quotient is 0 and the remainder is 2.) Note that the division continues until the quotient is zero. The answer is read from the bottom number to the top number in the remainder column. Thus, 328 2303 .5 = a) Convert 683 to base 5 by this method. 102135 b) Convert 763 to base 8 by this method. 13738 Concept/Writing Exercises 62. a) What is 04 equal to in base 10? 0 b) What is 14 equal to in base 10? 1 c) What is 24 equal to in base 10? 2 d) What is 34 equal to in base 10? 3 e) In general, if n is a digit less than the base b, and the base b is less than or equal to 10, then describe what nb is equal to in base 10. n 63. a) What is 102 equal to in base 10? 2 b) What is 108 equal to in base 10? 8 c) What is 1016 equal to in base 10? 16 d) What is 1032 equal to in base 10? 32 e) In general, for any base b, what is l0b equal to in base 10? b *See Instructor Answer Appendix Ildi Papp/Shutterstock
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