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1362 Number Systems (on CD) Appendix E positional value, and sum these products. For example, the binary number 110101 is converted to decimal 53 as shown in Fig. E.8. To convert octal 7614 to decimal 3980, we use the same technique, this time using appropriate octal positional values as shown in Fig. E.9. To convert hexadecimal AD3B to decimal 44347, we use the same technique, this time using appropriate hexadecimal positional values as shown in Fig. E.10. E.5 Converting from Decimal to Binary, Octal, or Hexadecimal The conversions of the previous section follow naturally from the positional notation conventions. Converting from decimal to binary, octal, or hexadecimal also follows these conventions. Suppose we wish to convert decimal 57 to binary. We begin by writing the positional values of the columns right to left until we reach a column whose positional value is greater than the decimal number. We do not need that column, so we discard it. Thus, we first write: Converting a binary number to decimal Positional values: 32 16 8 4 2 1 Symbol values: 1 1 0 1 0 1 Products: 1*32=32 1*16=16 0*8=0 1*4=4 0*2=0 1*1=1 Sum: = 32 + 16 + 0 + 4 + 0 + 1 = 53 Fig. E.8 Converting a binary number to decimal. Converting an octal number to decimal Positional values: 512 64 8 1 Symbol values: 7 6 1 4 Products 7*512=3584 6*64=384 1*8=8 4*1=4 Sum: = 3584 + 384 + 8 + 4 = 3980 Fig. E.9 Converting an octal number to decimal. Converting a hexadecimal number to decimal Positional values: 4096 256 16 1 Symbol values: A D 3 B Products A*4096=40960 D*256=3328 3*16=48 B*1=11 Sum: = 40960 + 3328 + 48 + 11 = 44347 Fig. E.10Fig. 10 Converting a hexadecimal number to decimal.
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Appendix E Number Systems (on CD) 1361 A particularly important relationship that both the octal number system and the hexadecimal number system have to the binary system is that the bases of octal and hexadecimal (8 and 16 respectively) are powers of the base of the binary number system (base 2). Consider the following 12-digit binary number and its octal and hexadecimal equivalents. See if you can determine how this relationship makes it convenient to abbreviate binary numbers in octal or hexadecimal. The answer follows the numbers. Binary Number Octal equivalent Hexadecimal equivalent 100011010001 4321 8D1 To see how the binary number converts easily to octal, simply break the 12-digit binary number into groups of three consecutive bits each, and write those groups over the corresponding digits of the octal number as follows 100 011 010 001 4321 Notice that the octal digit you have written under each group of thee bits corresponds precisely to the octal equivalent of that 3-digit binary number as shown in Fig. E.7. The same kind of relationship may be observed in converting numbers from binary to hexadecimal. In particular, break the 12-digit binary number into groups of four consecutive bits each and write those groups over the corresponding digits of the hexadecimal number as follows 1000 1101 0001 8D1 Notice that the hexadecimal digit you wrote under each group of four bits corresponds precisely to the hexadecimal equivalent of that 4-digit binary number as shown in Fig. E.7. E.3 Converting Octal Numbers and Hexadecimal Numbers toBinary Numbers In the previous section, we saw how to convert binary numbers to their octal and hexadecimal equivalents by forming groups of binary digits and simply rewriting these groups as their equivalent octal digit values or hexadecimal digit values. This process may be used in reverse to produce the binary equivalent of a given octal or hexadecimal number. For example, the octal number 653 is converted to binary simply by writing the 6 as its 3-digit binary equivalent 110, the 5 as its 3-digit binary equivalent 101, and the 3 as its 3digit binary equivalent 011 to form the 9-digit binary number 110101011. The hexadecimal number FAD5 is converted to binary simply by writing the F as its 4-digit binary equivalent 1111, the A as its 4-digit binary equivalent 1010, the D as its 4digit binary equivalent 1101, and the 5 as its 4-digit binary equivalent 0101 to form the 16digit 1111101011010101. E.4 Converting from Binary, Octal, or Hexadecimal to Decimal Because we are accustomed to working in decimal, it is often convenient to convert a binary, octal, or hexadecimal number to decimal to get a sense of what the number is really worth. Our diagrams in Section E.1 express the positional values in decimal. To convert a number to decimal from another base, multiply the decimal equivalent of each digit by its
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1360 Number Systems (on CD) Appendix E Positional values in the hexadecimal number system Decimal digit 3 D A Position name Two-hundred-and-Sixteens Ones fifty-sixes Positional value 256 16 1 Positional value as a 162 161 160 power of the base (16) Fig. E.6Positional values in the hexadecimal number system. Fig. For longer hexadecimal numbers, the next positions to the left would be the four-thousand-and-ninety-sixes position (16 to the 3rd power), the sixty-five-thousand-five-hundred-and-thirty-six position (16 to the 4th power), and so on. E.2 Abbreviating Binary Numbers as Octal Numbers and Hexadecimal Numbers The main use for octal and hexadecimal numbers in computing is for abbreviating lengthy binary representations. Figure E.7 highlights the fact that lengthy binary numbers can be expressed concisely in number systems with higher bases than the binary number system. Decimal number Binary{ representation Octal representation Hexadecimal representation 0 0 0 0 1 1 1 1 2 10 2 2 3 11 3 3 4 100 4 4 5 101 5 5 6 110 6 6 7 111 7 7 8 1000 10 8 9 1001 11 9 10 101012 A 11 101113 B 12 110014 C 13 110115 D 14 111016 E 15 111117 F 16 10000 20 10 Fig. E.7Decimal, binary, octal, and hexadecimal equivalents. Fig.
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Appendix E Number Systems (on CD) 1359 For longer decimal numbers, the next positions to the left would be the thousands position (10 to the 3rd power), the ten-thousands position (10 to the 4th power), the hundred- thousands position (10 to the 5th power), the millions position (10 to the 6th power), the ten-millions position (10 to the 7th power), and so on. In the binary number 101, we say that the rightmost 1 is written in the ones position, the 0 is written in the twos position, and the leftmost 1 is written in the fours position. Notice that each of these positions is a power of the base (base 2), and that these powers begin at 0 and increase by 1 as we move left in the number (Fig E.4). For longer binary numbers, the next positions to the left would be the eights position (2 to the 3rd power), the sixteens position (2 to the 4th power), the thirty-twos position (2 to the 5th power), the sixty-fours position (2 to the 6th power), and so on. In the octal number 425, we say that the 5 is written in the ones position, the 2 is written in the eights position, and the 4 is written in the sixty-fours position. Notice that each of these positions is a power of the base (base , and that these powers begin at 0 and increase by 1 as we move left in the number (Fig. E.5). For longer octal numbers, the next positions to the left would be the five-hundred-andtwelves position (8 to the 3rd power), the four-thousand-and-ninety-sixes position (8 to the 4th power), the thirty-two-thousand-seven-hundred-and-sixty eights position (8 to the 5th power), and so on. In the hexadecimal number 3DA, we say that the A is written in the ones position, the D is written in the sixteens position, and the 3 is written in the two-hundred-and-fifty-sixes position. Notice that each of these positions is a power of the base (base 16), and that these powers begin at 0 and increase by 1 as we move left in the number (Fig. E.6). Positional values in the binary number system Binary digit 1 0 1 Position name Fours Twos Ones Positional value 4 2 1 Positional value as a 22 21 20 power of the base (2) Fig. E.4Fig. Positional values in the binary number system. Positional values in the octal number system Decimal digit 4 2 5 Position name Sixty-fours Eights Ones Positional value 64 8 1 Positional value as a 82 81 80 power of the base (8) Fig. E.5Fig. Positional values in the octal number system.
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1358 Number Systems (on CD) Appendix E Notice that each of these positions is a power of the base (base 10), and that these powers begin at 0 and increase by 1 as we move left in the number (Fig. E.3). Binary digit Octal digit Decimal digit Hexadecimal digit 0 0 0 0 1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 9 9 A (decimal value of 10) B (decimal value of 11) C (decimal value of 12) D (decimal value of 13) E (decimal value of 14) F (decimal value of 15) Fig. E.1Fig. Digits of the binary, octal, decimal and hexadecimal number systems. Attribute Binary Octal Decimal Hexadecimal Base 2 8 10 16 Lowest digit 0000 Highest digit 179F Fig. E.2Comparing the binary, octal, decimal and hexadecimal number systems. Fig. E.2 Positional values in the decimal number system Decimal digit 937 Position name Hundreds Tens Ones Positional value 10010 1 Positional value as a 102 101 100 power of the base (10) Fig. E.3Positional values in the decimal number system. Fig.
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Appendix E Number Systems (on CD) 1357 Outline E.1 Introduction E.2 Abbreviating Binary Numbers as Octal Numbers and Hexadecimal Numbers E.3 Converting Octal Numbers and Hexadecimal Numbers to Binary Numbers E.4 Converting from Binary, Octal, or Hexadecimal to Decimal E.5 Converting from Decimal to Binary, Octal, or Hexadecimal E.6 Negative Binary Numbers: Two s Complement Notation Summary Terminology Self-Review Exercises Answers to Self-Review Exercises Exercises E.1 Introduction In this appendix, we introduce the key number systems that Java programmers use, especially when they are working on software projects that require close interaction with machine-level hardware. Projects like this include operating systems, computer networking software, compilers, database systems, and applications requiring high performance. When we write an integer such as 227 or -63 in a Java program, the number is assumed to be in the decimal (base 10) number system. The digits in the decimal number system are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The lowest digit is 0 and the highest digit is 9 one less than the base of 10. Internally, computers use the binary (base 2) number system. The binary number system has only two digits, namely 0 and 1. Its lowest digit is 0 and its highest digit is 1 one less than the base of 2. As we will see, binary numbers tend to be much longer than their decimal equivalents. Programmers who work in assembly languages and in high-level languages like Java that enable programmers to reach down to the machine level, find it cumbersome to work with binary numbers. So two other number systems the octal number system (base and the hexadecimal number system (base 16) are popular primarily because they make it convenient to abbreviate binary numbers. In the octal number system, the digits range from 0 to 7. Because both the binary number system and the octal number system have fewer digits than the decimal number system, their digits are the same as the corresponding digits in decimal. The hexadecimal number system poses a problem because it requires sixteen digits a lowest digit of 0 and a highest digit with a value equivalent to decimal 15 (one less than the base of 16). By convention, we use the letters A through F to represent the hexadecimal digits corresponding to decimal values 10 through 15. Thus in hexadecimal we can have numbers like 876 consisting solely of decimal-like digits, numbers like 8A55F consisting of digits and letters, and numbers like FFE consisting solely of letters. Occasionally, a hexadecimal number spells a common word such as FACE or FEED this can appear strange to programmers accustomed to working with numbers. Each of these number systems uses positional notation each position in which a digit is written has a different positional value. For example, in the decimal number 937 (the 9, the 3, and the 7 are referred to as symbol values), we say that the 7 is written in the ones position, the 3 is written in the tens position, and the 9 is written in the hundreds position.
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E Number Systems (on CD) Objectives To understand basic number systems concepts such as base, positional value, and symbol value. To understand how to work with numbers represented in the binary, octal, and hexadecimal number systems To be able to abbreviate binary numbers as octal numbers or hexadecimal numbers. To be able to convert octal numbers and hexadecimal numbers to binary numbers. To be able to covert back and forth between decimal numbers and their binary, octal, and hexadecimal equivalents. To understand binary arithmetic, and how negative binary numbers are represented using two s complement notation. Here are only numbers ratified. William Shakespeare Nature has some sort of arithmetic-geometrical coordinate system, because nature has all kinds of models. What we experience of nature is in models, and all of nature s models are so beautiful. It struck me that nature s system must be a real beauty, because in chemistry we find that the associations are always in beautiful whole numbers there are no fractions. Richard Buckminster Fuller
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D ASCII Character Set 0123456789 0 1 2 3 4 5 6 7 8 9 10 11 12 nul soh stx etx eot enq ack bel bs ht nl vt ff cr so si dle dc1 dc2 dc3 dc4 nak syn etb can em sub esc fs gs rs us sp ! ” # $ % & ( ) * + , -. / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ ] ^ _ a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~ del Fig. D.1ASCII character set. Fig. The digits at the left of the table are the left digits of the decimal equivalent (0-127) of the character code, and the digits at the top of the table are the right digits of the character code. For example, the character code for F is 70, and the character code for & is 38. Most users of this book are interested in the ASCII character set used to represent English characters on many computers. The ASCII character set is a subset of the Unicode character set used by Java to represent characters from most of the world s languages. For more information on the Unicode character set, see Appendix K.
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1354 Operator Precedence Chart Appendix C OperatorOperatorOperatorOperatorTypeTypeTypeTypeAssociativityAssociativityAssociativityAssociativity < <= > >= instanceof relational less than relational less than or equal to relational greater than relational greater than or equal to type comparison left to right == != relational is equal to relational is not equal to left to right & bitwise AND left to right ^ bitwise exclusive OR boolean logical exclusive OR left to right | bitwise inclusive OR boolean logical inclusive OR left to right && logical AND left to right || ?: = += -= *= /= %= &= ^= |= <<= >>= >>>= logical OR ternary conditional assignment addition assignment subtraction assignment multiplication assignment division assignment modulus assignment bitwise AND assignment bitwise exclusive OR assignment bitwise inclusive OR assignment bitwise left shift assignment bitwise right shift with sign extension assibitwise right shift with zero extension assileft to right right to left gnment gnment right to left Fig. C.1Operator precedence chart (part 2 of 2). Fig.
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C Operator Precedence Chart Operators are shown in decreasing order of precedence from top to bottom. OperatorOperatorOperatorOperatorTypeTypeTypeTypeAssociativityAssociativityAssociativityAssociativity () [] . ++ ++ + - ! ~ ( type ) * / % + - << >> >>> parentheses array subscript member selection unary postincrement unary postdecrement unary preincrement unary predecrement unary plus unary minus unary logical negation unary bitwise complement unary cast multiplication division modulus addition subtraction bitwise left shift bitwise right shift with sign extension bitwise right shift with zero extension left to right right to left right to left left to right left to right left to right Fig. C.1Fig. Operator precedence chart (part 1 of 2).
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