The function
giving the fractional (noninteger) part of a real number . The symbol is sometimes used instead of (Graham et al. 1994, p. 70; Havil 2003,
p. 109), but this notation is not used in this work due to possible confusion
with the set containing the element .

Unfortunately, there is no universal agreement on the meaning of for and there are two common definitions. Let be the floor function,
then the Wolfram Language command
FractionalPart[x]
is defined as

(1)

(left figure). This definition has the benefit that , where is the integer part of
. Although Spanier and Oldham (1987) use
the same definition as the Wolfram Language,
they mention the formula only very briefly and then say it will not be used further.
Graham et al. (1994, p. 70), and perhaps most other mathematicians, use
the different definition

(2)

(right figure).

The fractional part function can also be extended to the complex
plane as

(3)

as illustrated above.

Since usage concerning fractional part/value and integer part/value can be confusing, the following table gives a summary of names and notations used. Here, S&O indicates Spanier and Oldham (1987).