Complements
         
         •
          Subtraction of numbers requires a different algorithm 
           than addition.
         •
          Adding a complement of a number is equivalent to 
         subtraction.
         •
          We will discuss two complements:
            Diminished Radix Complement
            Radix Complement
         •
          Subtraction will be accomplished by adding a 
                complement.
          SYEN 3330 Digital Systems                  Chapter 5-2       Page 
                                                               2
         Diminished Radix Complement
            
        Given a number   in Base   having   digits, the (                     )'s 
                               N             r           n                 r-1
        complement (called the Diminished Radix Complement) is 
        defined as: 
                         ( n       ) -    
                          r   -  1     N
        Example: 
                         For   = 10,   = 1234 ,   = 4  (4 digits),we have: 
                               r         N           10 n
                                   ( n -   ) = 10,000 -1 = 9999
                                    r     1                           10 
                         The 9's complement of 123410 is then: 
                                   999910 - 123410 = 876510 
             SYEN 3330 Digital Systems                                   Chapter 5-2       Page 
                                                                                       3
          Binary 1's Complement 
              
         For   = 2,   = 01110011 ,   = 8  (8 digits), we have: 
                r        N                     2 n
                                          ( n -   ) = 256 -1 = 255             or 11111111
                                         r      1                          10                      2 
         The 1's complement of 011100112 is then: 
                                          111111112 
                                          - 011100112  
                                      ---------------- 
                                          100011002 
            NOTE:  Since the 2n-1 factor consists of all 1's and 
                   since 1 - 0= 1 and 1 - 1 = 0, forming the one's 
                   complement consists of  complementing each  
                    individual bit . 
               SYEN 3330 Digital Systems                                              Chapter 5-2       Page 
                                                                                                       4
          Radix Complement
              
         Given a number   in Base   having   digits, the  's 
                                     N                r             n                    r
         complement (called the Radix Complement) is defined as: 
                                     n                        
                                    r   - N       for  N   0  and 
                                           0        for  N = 0 
      Note that the Radix Complement is obtained by adding 1 
      to the Diminished Radix Complement. 
          Example: 
          For r = 10, N = 123410, n = 4  (4 digits), we have: 
               n  = 10,000
              r                   10 
          The 10's complement of 123410 is then 
             10,000  - 1234  = 8766                      or  8765 + 1 (9's complement plus 1) 
                        10            10             10 
               SYEN 3330 Digital Systems                                                Chapter 5-2       Page 
                                                                                                          5
           Binary 2's Complement 
               
       For   = 2,   = 01110011 ,   = 8  (8 digits), we have: 
              r         N                      2 n
        
                               ( n ) = 256            or 100000000
                                r                10                         2 
        
       The 2's complement of 01110011  is then:
                                                               2               
                                    1000000002 
                                    - 01110011   
                                                       2
                                  ---------------- 
                                      100011012 
        Note that this is the 1's complement plus 1. 
                SYEN 3330 Digital Systems                                                    Chapter 5-2       Page 
                                                                                                                6