The next important point is that there is a
gap between , the smallest positive number, and 0.0.
A round-off
error in a calculation that should produce a small non-zero value
but instead results in 0.0 is called an * underflow*. One of the
strengths of the IEEE standard is that it allows a special
* denormalized* form for very small numbers in order to stave off
underflows as long as possible. This is why the exponent in the
largest and smallest positive numbers are not symmetrical.
Without denormalized numbers, the smallest positive number in the
IEEE standard would be around .

Finally, and perhaps most
important, is the fact that the numbers that can be represented
are not distributed evenly throughout the range. Representable
numbers are very dense close to 0.0, but then grow steadily
further apart as they increase in magnitude. The dark regions in
Figure 2
correspond to parts of the number line where
representable numbers are packed close together. It is easy to
see why the distribution is not even by asking what two numbers
are represented by two successive values of the mantissa for any
given exponent. To make the calculations easier, suppose we have
a 16-bit system with a 7-bit mantissa and 8-bit exponent. No
matter what the exponent is, the distance between any two
successive values of the mantissa, e.g. between
and , will be .
For numbers closest to 0.0, the exponent will be a negative
number, e.g. **-100**, and the distance between two successive floating
point numbers will be . At the other
end of the scale, when
exponents are large, the distance between two numbers will be
approximately , namely .