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					     Computer
   Fundamentals
Lecture 3 :Computer Arithmetic
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   Objectives
      After completing this module you will be able to:
         Perform basic arithmetic operations using
          binary number system
         Explain the use of special memory locations in
          CPU to keep track of the status of last
          operation
         Signed and Unsigned Integers
         Sign-Magnitude




Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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   Lecture Outline
      Binary Addition
      Binary Subtraction
      Number Representation
      Overflow & Underflow




Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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   Binary Addition
Rules of Binary Addition
 0 + 0 = 0
 0 + 1 = 1
 1 + 0 = 1
 1 + 1 = 0, and carry 1 to the next more significant bit
 1 + 1 + 1 = 1, and carry 1 to the next more significant bit




Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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    Binary Addition - Examples
     101               5                             00011010 +                     26
    + 101              +5                             00001100                       12

       1010                   10                      00100110                         38


                                                             Exercise
            00010011 + 19                                        10001 + 11101
            00111110 62




Computer Fundamentals (101)        Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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   Binary Addition - Examples
   1.     10001 + 11101 = ?
   2.     1110 + 1111 = ?
   3.     101101 + 11001 = ?
   4.     10111 + 110101 = ?
   5.     1011001 + 111010 = ?
   6.     11011 + 1001010 = ?




Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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   Binary Addition- Answers




Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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   Binary Subtraction
Rules of Binary Subtraction
 0 - 0 = 0
 0 - 1 = 1, and borrow 1 from the next more
             significant bit
 1 - 0 = 1
 1 - 1 = 0

   Just like subtraction in any other base
         Minuend                        10110
         Subtrahend                     -10010
         Difference                     00100

Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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    Binary Subtraction
   And when a borrow is needed. Note that the borrow
    gives us 2 in the current bit position.




   Try out these subtractions: Exercise: 2
        0 2
                                    00110011                                                    51
      00100101     37             - 00010110                                                   22
     - 00010001 - 17
       00010100      20             00011101                                                   29
Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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   Binary Subtraction(Contd.)




Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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Binary Subtraction - Examples
   1011011 − 10010 = ?
   100010110 − 1111010 = ?
   1010110 − 101010 = ?
   101101 − 100111 = ?
   1000101 − 101100 = ?
   1110110 − 1010111 = ?




Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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  Binary Subtraction - Answers




Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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     Overflow & Underflow
      Overflow
          Occurs when the result of an operation is too large to
           store in the location allocated.
          0 1100 12
          0 0110 06
          1 0010 18
      Underflow
            Occurs when the result of an operation is too small to
             store in the location allocated.




Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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   Binary Addition with overflow
      Please add the following two numbers (8 bit
       format)
         1001 0110 +
         1111 1001
       11000 1111

      Since we have only 8 bit format the result has no
       place to appear.
      Thus it generates an OVERFLOW.
      Unless we handle them we get wrong results.



Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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   Binary Addition with overflow
   (Contd.)
    In the CPU there are special places to hold
     these two important information.
    In computer processors the Carry flag
     (usually indicated as the C Flag/ C bit) is a
     single bit in a system status (flag) register
    It is used to indicate when an arithmetic
     carry or borrow has been generated out of
     the most significant bit position.



Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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   Binary Addition
    In computer processors, the oVerflow
     flag (sometimes called V flag/ V bit) is
     usually a single bit in a system status
     register
    It is used to indicate when an arithmetic
     overflow has occurred in an operation.
    These places in the CPU can be accessed
     through a program.
            This means that when we design a program we need to
             first have an understanding of the extent of the data
             format that we are going to handle.

Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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   Unsigned and Signed Integers
      Unsigned Integers : Unsigned integer is
              either positive or zero value of an
              integer.
            Example : The number of integers between
                       0 and + 127

      Signed Integer: signed integer is either
               negative value or positive value of
               the integer.
            Example : The number of integers between
                       - 128 and + 127

Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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   Unsigned and Signed integers (Contd.)




Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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                              Sign-Magnitude
      Sign-Magnitude Representation
          There are many schemes for representing
           negative integers with patterns of bits.
           One scheme is sign-magnitude. It uses one bit
           (usually the leftmost) to indicate the sign.
               "0“ - indicates a positive integer
               "1“ - indicates a negative integer


            The rest of the bits are used for the magnitude
           of    the number.
            Eg. -2410 is represented as: 1001 1000

Computer Fundamentals (101)     Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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                              Sign-Magnitude
    Sign-Magnitude Representation
   Sign Bit
     0000 1001          1000 1001
                 +9                                                -9

    Problem: two values for zero
     0000 0000              1000 0000
    8-bit number can represent from
   -127 (1111 1111) to +127 (0111 1111)

Computer Fundamentals (101)     Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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   Number Representation
    001011                    +01011
    001110                    +01110
                              +11001                               011001

   001011                     +01011
   100110                     -00110
                              +00101                               000101



Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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   Number Representation
   Addition
    001011                         +01011
    110110                    +    -10110
                                   -01011                               101011
   Subtraction
    001011                          +01011
    001001                        - +01001
                                    +00010                              000010


Computer Fundamentals (101)        Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology
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                              Thank You
                                Next Lecture

           Lecture 04: Complimentary Arithmetic




Computer Fundamentals (101)   Lecture 03 - Computer Arithmetic   © Sri Lanka Institute of Information Technology

				
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