188 CHAPTER 4 Systems of Numeration Have you done any of the following today: spent time on Facebook, Instagram, or Snapchat, shopped online, used your smartphone to send a text message or to listen to music, or received driving directions? These activities all involve using a device with a computer chip that operates with numerals other than our familiar base 10 (decimal) numerals. In this section, we will study different bases, including base 2 (binary), base 8 (octal), and base 16 (hexadecimal) numerals, all of which are used in most modern computers and electronic devices. Other Bases SECTION 4.3 LEARNING GOALS Upon completion of this section, you will be able to: 7 Convert between Hindu–Arabic numerals and numerals with bases less than ten. 7 Convert between Hindu–Arabic numerals and numerals with bases greater than ten. Why This Is Important In addition to learning about the mathematics used in designing and programming electronic devices, studying other bases helps us better understand how our own decimal system works. The positional values in the Hindu–Arabic numeration system are , (10) , (10) , (10) , 10, 1 4 3 2 … The positional values in the Babylonian numeration system are , (60) , (60) , (60) , 60, 1 4 3 2 … Thus, 10 and 60 are called the bases of the Hindu–Arabic and Babylonian systems, respectively. Any counting number greater than 1 may be used as a base for a positional-value numeration system. If a positional-value system has a base b, then its positional values will be b b b b ,,,,,1 4 3 2 … For example, the positional values in a base 8 system are ,8 ,8 ,8 ,8,1 4 3 2 … and the positional values in a base 2 system are ,2 ,2 ,2 ,2,1 4 3 2 … In Exercises 56–59, first convert each numeral to a Hindu– Arabic numeral and then perform the indicated operation. Finally, convert the answer back to a numeral in the original numeration system. 56. 1 * 57. 2 * 58. 1 * 59. 2 * 60. Your Own System Create your own place-value system. Write 2025 in your system. Answers will vary. Research Activities 61. Hindu–Arabic System Investigate and write a report on the development of the Hindu–Arabic system of numeration. Start with the earliest records of this system in India. 62. Eastern Arabic Numerals The Arabic numeration system currently in use in Iran, Afghanistan, Egypt, and Sudan, is a base 10 positional-value system, which uses different symbols than the Hindu–Arabic numeration system. Write the symbols used in the Eastern Arabic system of numeration and their equivalent symbols in the Hindu– Arabic numeration system. Write 54,607, and 2025 in Eastern Arabic numerals. *See Instructor Answer Appendix Africa Studio/Shutterstock
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