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clear that an irrational number such as is also represented most accurately
by a normalized representation: signi cand bits should not be wasted by
storing leading zeros. However, the number zero is special. It cannot be
normalized, since all the bits in its representation are zero. The exponent E
is irrelevant and can be set to zero. Thus, zero could be represented as
0 E = 0 0.0000000000000000000000 :
The gap between the number 1 and the next largest oating point number
is called the precision of the oating point system, 1 or, often, the machine
precision, and we shall denote this by . In the system just described, the
next oating point bigger than 1 is
with the last bit b22 = 1. Therefore, the precision is = 2;22.
Exercise 1 What is the smallest possible positive normalized oating point
number using the system just described?
Exercise 2 Could nonzero numbers instead be normalized so that 2 m <
1? Would this be just as good?
It is quite instructive to suppose that the computer word size is much
smaller than 32 bits and work out in detail what all the possible oating
numbers are in such a case...
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This note was uploaded on 02/12/2014 for the course MATH 4800 taught by Professor Lie during the Spring '09 term at Rensselaer Polytechnic Institute.
- Spring '09