Page 53 - Computer Science Class 11 Without Functions
P. 53

Example 21: (0.010011001) to decimal
                                    2
              Face Value       0.     0     1      0     0      1      1     0      0     1
              Place Value      2 0   2 –1   2 –2   2 –3   2 –4   2 –5   2 –6   2 –7   2 –8   2 –9

                               –1
                                                                                      –8
                                                                              –7
                                       –2
                                                               –5
                                                                       –6
                                               –3
                                                       –4
                         = 0 × 2  + 1 × 2 + 0 × 2 + 0 × 2 + 1 × 2 + 1 × 2 + 0 × 2 + 0 × 2 + 1 × 2 –9
                            1     1    1     1
                         =   +   +   +
                           2 2   2 5   2 6   2 9
                            1     1      1     1
                         =   +       +     +
                            4     32    64    512
                         = 0.25 + 0.03125 + 0.01562 + 0.001953
                         = (0.29882)
                                    10
            Note that the number (0.298828125)  is slightly less than (0.3) . Recall that the binary representation of (0.3)  being
                                             10
                                                                                                             10
                                                                    10
            non-terminating, we had to ignore the bits beyond a certain number of bits (nine bits in the above example). So, when
            we convert the truncated binary number to decimal, we do not get back the number (0.3) .
                                                                                            10
            Example 22: (1100101.01) to decimal
                                   2
              Face Value        1    1     0    0     1     0    1     .    0     1
              Place Value       2 6  2 5   2 4  2 3   2 2   2 1  2 0        2 –1  2 –2
                     6
                                                                0
                                   4
                            5
                                                         1
                                                                       –1
                                                 2
                                          3
               = 1 × 2  + 1 × 2  + 0 × 2  + 0 × 2  + 1 × 2  + 0 × 2  + 1 x 2  + 0 × 2  + 1 × 2 –2
               = 64 + 32 + 0 + 0 + 4 + 0 + 1 +   0   +   1
                                          2   4
               = (101.25) 10
            Conversion of Fractional Numbers from Decimal to Octal
            To convert fractional numbers in the decimal number system to octal, follow the following steps:
            1.  Multiply the fractional part by 8.
            2.  Note the integer part.
            3.  Multiply only the fractional part by 8.
            4.   In the case of terminating fractions, repeat steps 2 and 3 till the fractional part becomes zero. In the case of
                non-terminating fractions, repeat steps 2 and 3 until the desired number of bits after the radix point has been
                obtained. Examples 23 and 24 illustrate the conversion of terminating and non-terminating fractions.
            5.  The integer part from top to bottom forms the octal number.
            Example 23: Convert (0.125)  to its equivalent octal number.
                                     10
                                     Fraction part   Integer part

              0.125 ×  8  =     1.0      0.0             1
              0.0 ×  8      =     0.0    0.0             0
              (0.125)  = (0.1) 8
                     10
            Example 24: Convert (0.740234375)  to its equivalent octal number.
                                            10
                                                 Fraction part    Integer part
              0.740234375 ×  8   =   5.921875      0.921875            5

              0.921875  ×  8         =  7.375        0.375             7
              0.375   ×  8              =   3.0       0.0              3
              (0.740234375)  = (0.573)
                           10        8





                                                                            Number Systems and Encoding Schemes  51
   48   49   50   51   52   53   54   55   56   57   58