Fixedpoint Representation of Numbers

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Fixed-point Representation of Numbers
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Fixed Point Representation of Numbers

• Sign-and-magnitude representation
• Twos complement representation
• Twos complement binary arithmetic
• Excess code representation
• Binary fractions
• Fixed point representation of fractional numbers

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Sign-and-magnitude representation

• magnitude of a number
• the numerical value of the no. excluding its sign
• the sign-and-magnitude method
• the leftmost bit represent the sign of the no.
• the remaining bits represent the binary
  equivalent of the magnitude of the no.
• sign bit - the bit represents the sign of a no.

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ExampleExpress 3710 and -1410 in 8-bit binary
codes using the sign-and-magnitude method

• convert the magnitude of the nos. into binary
• 3710 1001012
• 1410 11102
• if necessary, add 0s at the left until 7 binary
  digits
• 3710 1001012 01001012
• 1410 11102 00011102
• add a 0 or 1 at the left according to the
  sign of the no.
• 3710 001001012
• - 1410 100011102

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Pros Cons of sign-and-magnitude method

• Advantages
• simple and easy to understand
• Disadvantages
• the no. zero can be represented in two ways
• 00000000 and 10000000
• the sign bit and the magnitude part have to be
  handled separately complicates the design of
  the circuit used for addition particularly

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Twos complement representation

• counters of some cassette recorders and most
  video cassette recorders (0000 - 9999)
• reset the counter to 0000
• rewind a tape
• the counter changes to 9999 decreases gradually
• similar to the design of the memory of a computer
  (00000000 - 11111111)

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Twos Compl.

• the leftmost bit of the binary code
• a ve no. ? 0
• a -ve no. ? 1
• the range of nos. represented is -128 to 127
• 11111111 1 00000000
• 01111111 1 10000000

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The relation between a no. its negative

• The codes 01111110 (126) 100000010 (-126) are
  called the twos complement of each other
• it can be obtained by
• changing all 0s to 1s and all 1s to 0s
• adding the resulting binary code by 1

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Example Obtain the twos complement of 00101110
and 10100010

• 00101110
• change to 11010001
• 11010001 1 11010010
• 10100010
• change to 01011101
• 01011101 1 01011110

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To convert a denary no. into an n-bit binary code
using the twos complement notation

• convert the magnitude of the denary no. into a
  binary no.
• add 0s at the left until it contains n digits
• if the original no. is negative, obtain the twos
  complement of the binary code
• (Note that we need not find the twos complement
  if the original no. is non-negative)

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Examplerepresent 3710 and -1410 in 8-bit twos
complement code

• 3710 1001012 001001012
• 1410 11102 000011102
• -1410 111100012 1 111100102

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To convert a 8-bit binary code back to its
denary equivalent

• if the leftmost bit is 0 (i.e. the no. is ve)
• treat it as a simple binary no. and convert it
  into denary
• if the leftmost bit is 1 (i.e. the no. is -ve)
• obtain its magnitude by finding its twos
  complement
• change it to a denary no. and add the sign - in
  front of it

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Advantages of twos complement representation

• electronic circuit used is very simple in design
• no separate circuit is needed for subtraction
• no separate circuit is needed to handle the sign
  bit and the magnitude part
• each no. is represented by a unique code
• avoid ambiguity and widens the range of no.
  represented

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Twos complement binary arithmetic

• Addition
• Subtraction
• Overflow
• all nos. are represented in twos complement
  notation, and the no. of bits of each binary code
  is fixed

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Addition

• addition is done in usual binary nos.
• if a carry occurs beyond the leftmost bit, it is
  discarded
• Example perform the following additions in twos
  complement binary arithmetic using 8-bit binary
  codes
• 60 41
• (-60) (-41)
• 60 (-41)

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60 41

• Check Denary equivalent
• 60
• ) 41
• 101

• Carry ? 111
• 00111100
• ) 00101001
• 01100101
• Ans. 01100101

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(-60) (-41)

• Check Denary equivalent
• -60
• ) -40
• -101

• carry ? 11 1
• 11000100
• ) 11010111
• 110011011
• The extra digit is discarded
• ans. 10011011

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60 (-41)

• Check Denary equivalent
• 60
• ) - 40
• 19

• Carry ? 111111
• 00111100
• ) 11010111
• 100010011
• The extra digit is discarded
• ans. 00010011

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Subtraction

• Since x - y x (-y),
• x - y is found by adding x to the twos
  complement of y.
• Example perform the following additions in twos
  complement binary arithmetic using 8-bit binary
  codes
• 60 - 41 60 (-41)
• 60 - (-41) 60 41
• (-60) - (-41) (-60) 41

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Overflow

• overflow is said to occur when at some stage
  during processing binary arithmetic, a no.
  outside the finite range is generated
• for 8-bit binary codes in twos complement form,
  the finite range is -12810 to 12710
• concentrate on the possible occurrence of
  overflow during addition only
• when two nos. to be added are of the same sign,
  overflow may occur

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To find 86 60

• Carry ? 11111
• 01010110
• ) 00111100
• 10010010

• Check Denary equivalent
• 86
• ) 60
• 146

The answer is 10010010, whose denary equivalent
is -110 i.e, overflow has occurred.
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Conclusion of Overflow

• In most microcomputers, 16-bit (or even 32-bit)
  binary codes are used
• the range becomes -32768 to 32767 inclusively
• However, no matter how many bits we use, it is
  still possible to find nos. outside the finite
  range

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Binary fractions

• denary nos.
• 1 2 . 6 2 5
• place value 101 100 10-1 10-2
  10-3
• decimal point
• binary fraction
• 1 0 . 0 1 1
• place value 21 20 2-1 2-2
  2-3
• binary point

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• 10.0112
• 21 2-2 2-3
• 2 1/4 1/8
• 2.37510

• 12.62510
• 12 0.625
• 12 5/8
• 12 1/2 1/8
• 11002 0.12 0.0012
• 1100.1012

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Fixed point representation of fractional numbers

• P. 87 - 88