About This Presentation
          This presentation is part of the “Math + Fun!” series devised 
          by Behrooz Parhami, Professor of Computer Engineering at 
          University of California, Santa Barbara. It was first prepared 
          for special lessons in mathematics at Goleta Family School 
          during three school years (2003-06). “Math + Fun!” material 
          can be used freely in teaching and other educational 
          settings. Unauthorized uses are strictly prohibited. © 
          Behrooz Parhami
              Edition       Released     Revised      Revised
              First         Nov. 2005
       Nov. 2005                  Math in Computers                Slide 2
                  Counters and Clocks
                                                 9   0   1
                                               8           2
                                               7           3
                                                 6   5   4
  
      Nov. 2005              Math in Computers           Slide 3
                   A Mechanical Calculator
    Photo of the 1874 
    hand-made version
    Photo of production version, 
    made in Sweden (ca. 1940)
     Odhner calculator: invented by Willgodt T. Odhner (Russia) in 1874
       Nov. 2005                   Math in Computers                  Slide 4
                             The Inside of an Odhner Calculator
                           1 9 7
              . . . 0 8 6 4 2
              +       5 3 6 5
                           14007
                       Nov. 2005                                                                                 Math in Computers                                                                                               Slide 5
                  Decimal versus Binary Calculator
                                                   5   0  2         5
                                                  1000   100  10     1
                                    0           5000 + no hundred + 20 + 5
                                                = Five thousand twenty-five
                                       1                                                          1      0     1      1
                                                                                                   8      4     2      1
                                        2                                                       8 + no 4 + 2 + 1 
                                                                                                      = Eleven
                                        3
                                                                                                 0
                                     4
        After movement by 10 notches                                After movement by 2 notches 
        (one revolution), move the next                             (one revolution), move the next 
        wheel to the left by 1 notch.                               wheel to the left by 1 notch. 
            Nov. 2005                                    Math in Computers                                       Slide 6