Section 21.3
Limitations

There are two limitations on floating point numbers which are represented in this or any similar way due to the finite space within a computer's memory and circuits:

  1. Magnitude limitations -- what is the largest absolute value that can be stored? Fixed by the number of digits in the exponent.
  2. Precision limitations -- what is the closest any two numbers can be in terms of size? Fixed by the number of digits in the mantissa.

Looking at the number line again, magnitude limitations place a boundary on how far to the left or right numbers can go, while the precision determines how far apart the representable numbers are from each other. Fig. 21.3.1 shows the regions of the real number line that can be represented based on the exponent.


Fig. 21.3.1: The number of bits in the exponent limits the range of magnitudes
that can be represented; only the shaded regions and 0 are representable

Fig. 21.3.2 shows that the number of bits in the mantissa determine how close two floating point numbers can be to one another. Though there are an infinite number of real numbers between them, not all are representable because there is no limit theoretically on the number of places after the decimal point. In any physical computer system, there must be due to finite space.


Fig. 21.3.2: The number of bits in the mantissa limits how close the representable
numbers are to each other; .073815 is not representable, for example.

Representation of real numbers as floating points is a matter of trade-offs and economics. When we need super precise numbers, we buy a different computer or write slower software. Most computers have at least two forms of floating point numbers: one short (single precision) and another long (double precision). Some computers even give you the choice of how large of an exponent to store. Many real-world and engineering applications do not require enormously precise values.

The IEEE 754 standard, used by many vendors and programming languages like Java, is a document that specifies a way to represent floating point numbers in a computer. Here are its limits given in decimal: 

                  # bits     # decimal
     type           total      digits            exponent range
-----------------------------------------------------------------
single precision     32          6          10-38 to 10+38
double precision     64         19          10-308 to 10+308