Page 57 - Cs_withBlue_J_C11_Flipbook
P. 57

Thus, a decimal number 12.5 can be represented as:
                 •  0.00125 × 10 4
                 •  0.125 × 10 2
                             3
                 •  0.0125 × 10  and so on
                 Here, we see that the exponent changes with the change in mantissa so that the value is not altered. Also, note the
                 change in the position of the decimal point.

                                                               Definition
                      The floating number representation of a number can implement high range of values. It consists of two parts namely
                      mantissa and exponent. The mantissa represents a signed fixed point number. The exponent denotes and designates
                      the position of the decimal (or binary) point.




                 Normalised Scientific Notation
                 In this notation, there is a single non-zero digit before the radix point.
                                                                                           2
                 A decimal number in 238.567 in normalised notation is represented as 2.38567 × 10 .
                 A binary number 110.011 in normalised notation will be 1.10011 × 2 . Since a binary number contains only 0 and 1, all
                                                                             2
                 the numbers will have a single 1 before the radix point in normalised representation.

                 Mantissa Exponent Notation
                 Floating point binary numbers are represented as Mantissa × 2 exponent  in this notation. Mantissa is represented in
                 normalised binary form.

                 Excess Notation
                 Excess N or biased notation is also used to represent signed floating point numbers. In this notation, a number called
                 magic number (N) is added to the original number to shift all the values by N. The number represented by this notation
                 is N less than the unsigned value.
                                                          Consider 3-bit numbers

                                            Numbers               Binary          Excess-4 notation
                                                0                  000                0 - 4 = -4
                                                1                  001                1 - 4 = -3
                                                2                  010                2 - 4 = -2
                                                3                  011                3 - 4 = -1
                                                4                  100                4 - 4 = 0
                                                5                  101                5 - 4 = 1
                                                6                  110                6 - 4 = 2
                                                7                  111                7 - 4 = 3

                 According to IEEE (Institute of Electrical and Electronics Engineers), a floating-point number has 3 parts which are as
                 follows:
                 •  The sign of mantissa: It assumes 0 for a positive number and 1 for negative.
                 •  The biased exponent: A bias (or excess) is added to the actual exponent to represent both positive and negative
                   exponents.
                 •  The mantissa: It represents the actual digits of floating point numbers. It contains a single leading bit before the radix
                   point and the fractional bits after the radix point. But the bits to the right of binary points are only stored internally.


                                                                                                                        55
                                                                                                            Encoding    55
   52   53   54   55   56   57   58   59   60   61   62