Chapter 1. Introduction
9
1.4.1
Trade offs between range and precision
It is crucial to point out that by using the floating point representation we have not
managed to represent more numbers. If for example we use a 64 bit floating point we
can still represent only 2
64
distinct numbers. However we have spread those distinct
representations to a wider range.
There will always be a compromise between the range and the desired precision. Since
there is only a given number of representation bits, by increasing the number of the
exponent bit we achieve a greater range however there is a significand loss of precision
as to which numbers in this range we can represent. Likewise by increasing the number
of bits in the significand part we increase the precision but there is a smaller range of
numbers to represent[
1.4.2
The floating point representation
A number representation specifies a way of storing a number in the form of a string of
digits. There are several mechanics by which strings of digits can represent numbers.
the digit string can be of any length, and the location of the radix point is indicated
by placing an explicit ”point” character (dot or comma) there. If the radix point is not
specified then it is implicitly assumed to lie at the right end of the string. In fixed-point
systems, some specific assumption is made about where the radix point is located in
the string. For example, the convention could be that the string consists of 8 decimal
digits with the decimal point in the middle, so that ”00012345” has a value of 1.2345.
In scientific notation, the given number is scaled by a power of 10 so that it lies within a
certain range—typically between 1 and 10, with the radix point appearing immediately
after the first digit. The scaling factor, as a power of ten, is then indicated separately
at the end of the number.
The floating point representation, as shown in
, consists of three parts.
• A signed string of digits of a given length, that represent the base of the number.
It is usually referred to as the significand or mantissa. The length of this string
determines the precision of the representation to be implemented. For example a