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Arithmetic with binary numbers

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• Adding binary numbers:
  • Addition rules:

         0     1     0     1
       + 0   + 0   + 1   + 1
       ---   ---   ---   ---
         0     1     1    10
                          ^
                          |
                          +--- carry
    

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  • Small unsigned example:

      
       In decimal     In binary
    
    		        ***      * = indicate that previous
            5           00000101         bit addition produced a carry
          + 7         + 00000111
         ----         ----------
           12           00001100
    

  • Large unsigned example:

       In decimal     In binary
    
                         **   *       * = indicate that previous bit addition
          145           10010001          produced a carry
        +  61         + 00111101
       -------        ----------
          206           11001110
    

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• Subtracting binary numbers:
  • Subtraction rules:

         0     1     0     1
       - 0   - 0   - 1   - 1
       ---   ---   ---   ---
         0     1    *1     0
                    ^
                    |
                    +--- BORROW !
    

  • When you borrow, you receive 2 more units (because digit values increases by 2 each time you move to the left one position).

  • Smale example:

       In decimal     In binary
    
                          *         * = indicate that previous
            9         00001001          bit subtraction produced
          - 5       - 00000101          a borrow
         ----       ----------
            4         00000100
    

  • Large example:

       In decimal     In binary
    
                       ** *             * = indicate that previous bit subtraction
            149        10010101             produced a borrow
          -  41      - 00101001
          -----      ----------
            108        01101100
    

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• Multiplying binary numbers:
  • Multiplication rules:

         0     1     0     1
       x 0   x 0   x 1   x 1
       ---   ---   ---   ---
         0     0     0     1
    

  • Small unsigned example:

             5            00000101
           x 3          x 00000011
          ----         -----------
    	15            00000101
    		     00000101*
    		   -----------
    		      00001111  = 15
    

  • Large unsigned example:

                 10010101    (= 149 dec)
               x 00101001    (= 41  dec)
               ----------
                 10010101
                00000000*
               00000000**
              10010101***
             00000000****
            10010101*****
           00000000******
          00000000*******
         ----------------
          001011111011101     (= 6109 = 149*41) 
    

  • NOTE: when you multiply 2 binary numbers of n bits, the result may be as large as 2n bits !

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• Dividing binary numbers:
  • Division rule:
    • Find number that is larger than divider
    • It will always devide once (much easier than division in decimal number system)

  • Small example:

        In decimal:                    In binary:
    
             03 (quotient)                  00000011 (quotient)
           -------                         ------------------
        3 /  10                        11 / 00001010
              9                                  11
            ----                                ---
              1 (remainder)                      100
    					      11
    					     ---
    					       1 (remainder)
    

  • Large example: 2237 divide by 27

       In decimal:
    
                        0082        (quotient = 82(10))
                      ---------
                  27 /  2237
                        0
                      ---
                        22
                         0
                       ---
                        223
                        216
                       ----
                          77
                          54
                         ---
                          23      (remainder = 23(10))
    
         In binary:
                       27(10) = 11011(2)
                     2237(10) = 100010111101(2)
    
                      000001010010    (quotient = 82(10))
                    -------------------
              11011/  100010111101
                      0
                    ---
                      10
                      00
                     ---
                      100
                      000
                     ----
                      1000
                      0000
                     -----
                      10001
                      00000
                     ------
                      100010
                       11011
                     -------
                         1111
                         0000
                      -------
                         11111
                         11011
                       -------
                           1001
    		       0000
                         ------
                           10011
                           00000
                          ------
                           100110
                            11011
                          -------
                             10111
                             00000
                            ------
                             10111    (remainder = 23(10))