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Information in a digital computer is represented by strings of digits that assume one of the two values 0 or 1 These units of information are called bits The term bit is a contraction for binary digit Bits are used internally in the computer to represent both numerical and non-numerical information In order to achieve ef ciency in dealing with information, a xed number of bits are grouped together and referenced as a discrete unit These units are used to encode and format the information that can be processed by the computer Units of 8 bits are common and are called bytes (Figure 325) Bytes are used to encode the basic symbolic characters that provide the computer with input-output information such as the alphabet, decimal digits, punctuation marks, and special characters The byte size is not universally adopted on all machines but is perhaps the most popular basic unit of information Larger bit groups are organized into units called words (and sometimes into smaller units called half words or larger units called double words) These units are used to encode the basic instruction repertoire of the machine and format the
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FIGURE 325 Illustration of principle of operation
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numerical data Common word sizes used on various commercially available computers are 16, 24, 32, 36, 48, and 60 bits Numerical information processed by a computer is represented in the binary numbering system The binary numbering system uses a position notation analogous to that used in decimal numbers For example, the decimal number 59637 repre3 10 1 7 10 2 The value sents the value 5 102 9 101 6 10 assigned to any of the 10 possible digits 0 through 9 depends on its position relative to the decimal point; ie, zero and positive exponents of 10 are assigned to digits appearing to the left of the decimal point and negative exponents of 10 to the right 23 0 22 In a similar manner, the binary number 1011011 represents 1 1 1 2 3 1 2 1 2 0 2 1 2 1 2 The radix (base) of this system is 2 Each position of a binary number is occupied by one of the two possible digits 0 or 1 The radix point in this system serves a purpose similar to that of the decimal point in the decimal system The functional operators available in the computer for setting up the solution of a problem are encoded into the words of the machine To handle numerical calculations, the instruction repertoire usually includes the four rules of arithmetic, ie, , , , and These instructions operate on data encoded in the binary system This, however, is not a serious operational problem, since the user speci es information for the problem in the decimal system, or through mnemonics, and the computer is used to convert these formats into its own internal representations
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Three different formats are used to represent numerical information internal to the computer: xed-point, oating-point and encoded-decimal These formats are described in this section a) A word in xed-point format is given as a string of 0 s and 1 s representing a binary number The radix point is not explicitly given but is implied by the program processing the data to be at a xed position in the word, eg, immediately to the right of the word so that the number represented would be an integer To represent algebraic numbers, several alternate forms are used Most often, 1 s or 2 s complement forms are adopted because they lead to a reduction and simpli cation of the electronics needed to perform the arithmetic operations b) Floating-point format is a mechanized version of the scienti c notation, ie, M 10 E, where M and E represent the signed mantissa and signed exponent of the number, respectively Using a machine word to represent numbers in this format, a large range of numbers is possible The signed mantissa
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