4.3 Other Bases 189 As we indicated in Section 4.2, the Mayan numeration system is based on the number 20. It is not, however, a true base 20 positional-value system. Why not? The reason for the almost universal acceptance of the base 10 numeration system is that most human beings have 10 fingers. However, throughout history, some societies have used numeration systems that use a base other than 10. Two such systems are the base 12, or duodecimal, system and the base 60, or sexagesimal system that we studied in Section 4.2. Our present-day society still contains remnants of these other base systems. For example, there are 12 inches in a foot, 12 months in a year, 12 items in a dozen, and 12 numbers on a clock. Furthermore, each hour contains 60 minutes and each minute contains 60 seconds. When measuring angles, 1 degree contains 60 minutes and each minute contains 60 seconds. Computers and many other electronic devices make use of three numeration systems: the binary (base 2), the octal (base 8), and the hexadecimal (base 16) numeration systems. One familiar example of the binary numeration system is the bar codes found on most items purchased in stores today. Computers can use the binary numeration system because the binary number system consists of only the digits 0 and 1. These digits can be represented with electronic switches that are either off (0) or on (1). All data that we enter into a computer can be converted into a series of single binary digits. Each binary digit is known as a bit. The octal numeration system is used when eight bits of data are grouped together to form a byte. In the American Standard Code for Information Interchange (ASCII) code, the byte 01000001 represents the character A, and 01100001 represents the character a. Other characters along with their decimal, binary, octal, and hexadecimal representation can be found at the website ASCIITable.com. The hexadecimal numeration system is used to create computer languages. Examples of computer languages that rely on the hexadecimal system are HTML, JavaScript, and CSS, all of which are used heavily in creating Internet web pages. Computers can easily convert between binary (base 2), octal (base 8), and hexadecimal (base 16) numbers. Bases Less Than 10 A place-value system with base b must have b distinct symbols, a symbol for zero and a symbol for each numeral less than the base. For example, a base 6 system must have symbols for 0, 1, 2, 3, 4, and 5. All numerals in base 6 are constructed from these 6 symbols. A base 8 system must have symbols for 0, 1, 2, 3, 4, 5, 6, and 7. All numerals in base 8 are constructed from these 8 symbols, and so on. A numeral in a base other than base 10 will be indicated by a subscript to the right of the numeral. Thus, 1235 represents a base 5 numeral. The numeral 1236 represents a base 6 numeral. The value of 1235 is not the same as the value of 123 , 10 and the value of 1236 is not the same as the value of 123 . 10 A base 10 numeral may be written without a subscript. For example, 123 means 12310 and 456 means 456 . 10 For clarity in certain problems, we will use the subscript 10 to indicate a numeral in base 10. The symbols that represent the base itself, in any base b, are 10.b For example, in base 5, the symbols 105 represent the number five. Note that 10 (15)(01) 50 5, 5 10 = × + × = + = or the number five in base 10. The numeral 105 means one group of 5 and no units. In base 6, the symbols 106 represent the number six. The symbols 106 represent one group of 6 and no units, and so on. To change a numeral in a base other than 10 to a base 10 numeral, we follow the same procedure we used in Section 4.2 to change the Babylonian and Mayan numerals to base 10 numerals. Multiply each digit by its respective positional value. Then determine the sum of the products. Lisa S./Shutterstock m Computers make use of the binary, octal, and hexadecimal number systems. Learning Catalytics Keyword: Angel-SOM-4.3 (See Preface for additional details.)
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