A floating-point number is represented by three quantities: the sign, the mantissa, and the exponent:
with 
and 
.
is called the mantissa, 
the basis, and e the exponent, with 
. 
is called the mantissa length. The condition 
makes the representation unique and saves, in the binary case (
), one bit.
Two-floating point zeros, 
and 
, exist, both represented by the mantissa 
.
On a typical Intel processor, 
. To represent a number in the float type, 64 bits are used, namely, 1 bit for the sign, 
bits for the mantissa, and 
bits for the exponent 
. The upper bound 
for the exponent is consequently 
.
With this data, the smallest positive representable number is
, and the largest 
.
Note that floating-point numbers are not equally spaced in 
. There is, in particular, a gap at zero (see also [29]). The distance between 
and the first positive number is 
, while the distance between the first and the second is smaller by a factor 
. This effect, caused by the normalization 
, is visualized in Figure 2.1:
This gap is filled equidistantly with subnormal floating-point numbers to which such a result is rounded. Subnormal floating-point numbers have the smallest possible exponent and do not follow the normalization convention, .