Converting Decimals to Octal and Hexadecimal Numbers in Java

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Converting Decimals to Octal and Hexadecimal Numbers
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Analogously, we can apply the above procedure for converting an octal to a binary The conversion for the decimal number 90 can be done as follows: 9010 = 11 8 + 2 1110 = 1 8 + 3 110 = 0 8 + 1 9010 = 1328 = 0132 The remainder values represent the digits in the equivalent octal number: 1328 This can be verified by back substitution, which gives the following result: 9010 = ((0 8 + 1) 8 + 3) 8 + 2 = 1 82 + 3 81 + 2 80 = 1328 = 0132 Conversion to hexadecimal is analogous: 9010 = 5 16 + 10 510 = 0 16 + 5 9010 = 5a16 = 0x5a The remainders represent the digits of the number in the hexadecimal system: 5a Back substitution gives the same result: 9010 = (0 16+ 5) 16 +10 = 5 161 + a 160 = 5a16 = 0x5a
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APPENDIX G
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G4 Representing Integers
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Integer data types in Java represent signed integer values, ie, both positive and negative integer values The values of char type can effectively be regarded as unsigned 16-bit integers Values of type byte are represented as shown in Table G2 A value of type byte requires 8 bits With 8 bits, we can represent 28 or 256 values Java uses 2 s complement (explained later) to store signed values of integer data types For the byte data type, this means values are in the range 128 ( 27) to +127 (27 1), inclusive Bits in an integral value are usually numbered from right to left, starting with the least significant bit 0 (also called the right-most bit) When applying bitwise operators, the number of the most significant bit (also called the left-most bit) is dependent on the integral type; bit 31 for byte, short, char, and int, and bit 63 for long The representation of the signed types sets the most significant bit to 1, indicating negative values Adding 1 to the maximum int value 2147483647 results in the minimum value -2147483648, ie, the values wrap around for integers and no overflow or underflow is indicated
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Table G2
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Representing Signed byte Values Using 2 s Complement Binary Representation (8 bit)
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01111111 01111110 00101001 00000010 00000001 00000000 11111111 11111110 11010111 10000001 10000000
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Decimal Value 127 126 41 2 1 0 -1 -2 -41 127 128
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Octal Value with Prefix 0
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0177 0176 051 02 01 00 0377 0376 0327 0201 0200
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Hexadecimal Value with Prefix 0x
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0x7f 0x7e 0x29 0x02 0x01 0x0 0xff 0xfe 0xd7 0x81 0x80
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APPENDIX G
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Calculating 2 s Complement
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Before we look at 2 s complement, we need to understand 1 s complement 1 s complement of a binary integer is computed by inverting the bits in the number Thus, 1 s complement of the binary number 00101001 is 11010110 1 s complement of a binary number N2 is denoted as ~N2 The following relations hold between a binary integer N2, its 1 s complement ~N2, and its 2 s complement N2: N2 = ~N2 + 1 0 = N2 + N2 If N2 is a positive binary integer, then N2 denotes its negative binary value, and vice versa The second relation states that adding a binary integer N2 to its 2 s complement N2 equals 0 Given a positive byte value, say 41, the binary representation of -41 can be found as follows: Binary Representation Given a value, N2: Form 1 s complement, ~N2: Add 1: Result is 2 s complement, N2:
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00101001 11010110 00000001 11010111
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Decimal Value 41
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Similarly, given a negative number, say -41, we can find the binary representation of 41: Binary Representation Given a value, N2: Form 1 s complement, ~N2: Add 1: Result is 2 s complement, -N2:
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11010111 00101000 00000001 00101001
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Decimal Value 41
APPENDIX G
Adding a number N2 to its 2 s complement N2 gives 0, and the carry bit from the addition of the most significant bits (after any necessary extension of the operands) is ignored: Binary representation Given a value, N2: Add 2 s complement, ~N2: Sum:
00101001 11010111 00000000