In the IEEE 754-2008 standard (referred to as IEEE 754 henceforth), a floating-point representation is an unencoded member of a floating-point format which represents either a finite number, a signed infinity, or some kind of NaN. An element of the subset of floating-point representations consisting of finite numbers and signed infinities is called a floating-point number.
A floating-point representation of a finite real number has three components: A sign, an exponent, and a significand.
The numerical value of a representation of a finite floating-point number is the
signed product of its significand and its radix 
raised to the power of its exponent; in particular,
note that the floating-point representation of a given value may not be unique, particularly
when the radix is 10 (IEEE Computer Society 2008).
When implemented at the machine-level, representations of floating-point data are encoded as 
-element 
-ary sequences of digits - here, 
is a parameter determined by the precision 
of the representation - which are partitioned into three distinct fields. Due to
the fact that values may have more than one representation, it may happen that encodings
fail to be in one-to-one correspondence with the values they represent. Even so,
IEEE 754 requires that all encoding procedures adhere to a strict set of guidelines
in terms of how each representation is encoded, thereby guaranteeing that both the
representation 
and the value 
of each floating-point datum be inferable from the fields
of the encoding itself. What's more, the partitions of any encoding are formatted
so that "auxiliary information" such as the payload
of a NaN can be deduced by decoding such an encoded representation
(IEEE Computer Society 2008, pp. 9-12).
