Floating-Point Representation




The  floating-point  representation  of  a  number  has  two  parts.  The  first  part represents  a  signed,  fixed-point  number  called  the  mantissa.  The  second  part designates  the  position  of  the  decimal  (or  binary)  point  and  is  called  the exponent.  The  fixed-point  mantissa  may  be  a  fraction  or  an  integer.  For  exam ple,  the  decimal  number  +6132.789  is  represented  in  floating-point  with  a fraction  and  an  exponent  as  follows: 

Fraction

+0.6132789 

Exponent +04 

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Number System Register Transfer Language -2
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Arithmetic Addition Instruction Codes
Flow Control Common Bus System-memory Address
Decimal Fixed-point Representation Memory-reference Instructions

The  value  of the  exponent  indicates  that the  actual  position  of the  decimal  point is  four  positions  to  the  right  of  the  indicated  decimal  point  in  the  fraction.  This representation  is  equivalent  to  the  scientific  notation  +0.6132789  X  10+4. Floating-point  is  always  interpreted  to  represent  a  number  in  the  follow ing  form: 

mxr' 

Only  the  mantissa  m  and  the  exponent  e  are  physically  represented  in  the register  (including  their  signs).  The  radix  r and  the  radix-point  position  of  the mantissa  are  always  assumed.  The  circuits  that  manipulate  the  floating-point numbers  in  registers  conform  with  these  two  assumptions  in  order  to  provide the  correct  computational  results. A  floating-point  binary  number is  represented  in  a  similar  manner  except that  it  uses  base  2  for  the  exponent.  For  example,  the  binary  number  + 1001.11 is  represented  with  an 8-bit  fraction  and  6-bit  exponent  as  follows: 

The fraction has a  0  in  the leftmost position to denote positive.  The binary point ofthe  fraction  follows  the sign bit but is  not  shown  in  the register.  The exponent has  the  equivalent  binary  number  +4.  The  floating-point  number  is  equivalent to 

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m  x  2'  =  +(.1001110),  x  z+• 



Frequently Asked Questions

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Ans: The representation of decimal numbers in registers is a function of the binary code used to represent a decimal digit. A 4-bit decimal code requires four flip-flops for each decimal digit. view more..
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Ans: An overflow condition can be detected by observing the carry into the sign bit position and the carry out of the sign bit position. If these two carries are not equal, an overflow condition is produced. view more..
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Ans: When two numbers of n digits each are added and the sum occupies n + 1 digits, we say that an overflow occurred. When the addition is performed with paper and pencil, an overflow is not a problem since there is no limit to the width of the page to write down the sum. view more..
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Ans: The floating-point representation of a number has two parts. The first part represents a signed, fixed-point number called the mantissa. The second part designates the position of the decimal (or binary) point and is called the exponent. The fixed-point mantissa may be a fraction or an integer. For exam ple, the decimal number +6132.789 is represented in floating-point with a fraction and an exponent as follows: view more..
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Ans: A floating-point number is said to be normalized if the most significant digit of the mantissa is nonzero. For example, the decimal number 350 is normalized but 00035 is not. Regardless of where the position of the radix point is assumed to be in the mantissa, the number is normalized only if its leftmost digit is nonzero. view more..
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Ans: In previous sections we introduced the most common types of binary-coded data found in digital computers. Other binary codes for decimal numbers and alphanumeric characters are sometimes used. Digital computers also employ other binary codes for special applications. A few additional binary codes encountered in digital computers are presented in this section. view more..
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Ans: Binary codes for decimal digits require a minimum of four bits. Numerous different codes can be formulated by arranging four or more bits in 10 distinct possible combinations. A few possibilities are shown in Table 3-6. view more..
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Ans: The ASCII code (Table 3-4) is the standard code commonly used for the transmission of binary information. Each character is represented by a 7-bit code and usually an eighth bit is inserted for parity (see Sec. 3-6). The code consists of 128 characters. Ninety-five characters represent graphic symbols that include upper- and lowercase letters, numerals zero to nine, punctuation marks, and special symbols view more..
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Ans: Binary information transmitted through some form of communication medium is subject to external noise that could change bits from 1 to 0, and vice versa. An error detection code is a binary code that detects digital errors during transmission. The detected errors cannot be corrected but their presence is indicated. The usual procedure is to observe the frequency of errors. If errors occur infrequently at random, the particular erroneous information is transmitted again. If the error occurs too often, the system is checked for malfunction view more..
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Ans: Parity generator and checker networl<s are logic circuits constructed with exclusive-OR functions. This is because, as mentioned in Sec. 1·2, the exclusive-OR function of three or more varia.bles is by definition an odd function. An odd function is a logic function whose value is binary 1 if, and only if, an odd function number of variables are equal to 1. According to this definition, the P( even) is the exclusive-OR of x, y, and l because it is equal to 1 when either one or all three of the variables are equal to I (Table 3-7). The P(odd) function is the complement of the P(even) function. view more..
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Ans: A digital system Is an interconnection of digital hardware module. that at'ClOinpl.lsh a specific Wormation-proceaslna taslc. Digital systems vary in size and complexi.ty interacting digital &om a few integrated circuits to a complex of interconnected and computers. Digital system design invariably UBeS a modular approach. The modules are constructed &om such digital components as ules registet&, are in decoders, terconnected arithmetic with common elements data and control paths , and control logic. The to fonn various moda digital computer system. view more..
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Ans: The symbolic notation used to describe the microoperation transfers among registers is called a register transfer language. The term "register transfer" implies the availability of hardware logic circuits that can perform a stated microoperation and transfer the result of the operation to the same or another register. view more..
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Ans: Computer registers are designated by capital letters (sometimes followed by numerals) to denote the function of the register. For example, the register that holds an address for the memory unit is usually called a memory address register and is designated by the name MAR. view more..
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Ans: where P is a control signal generated in the control section. It is sometimes convenient to separate the control variables from the register transfer operation by specifying a control function. view more..
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Ans: A typical digital computer has many registers, and paths must be provided to transfer information from one register to another. The number of wires will be excessive if separate lines are used between each register and all other registers in the system. view more..
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Ans: The two selection lines S1 and S0 are connected to the selection inputs of all four multiplexers. The selection lines choose the four bits of one register and transfer them into the four-line common bus. When S1S0 = 00, the 0 data inputs of all four multiplexers are selected and applied to the outputs that form the bus view more..
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Ans: A bus system can be constructed with three-state gates instead of multiplexers. A three-state gate is a digital circuit that exhibits three states. Two of the states are signals equivalent to logic 1 and 0 as in a conventional gate. The third state is a high-impedance state. view more..
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Ans: The operation of a memory unit was described in Sec. 2-7. The transfer of information from a memory word to the outside environment is called a read operation. view more..




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