Fractional Part

The fractional part function gives the 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 and the ceiling function. 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).

notation	name	S&O	Graham et al.	Wolfram Language
	ceiling function	--	ceiling, least integer	Ceiling[x]
	congruence	--	--	Mod[m, n]
	floor function		floor, greatest integer, integer part	Floor[x]
	fractional value		fractional part or	SawtoothWave[x]
	fractional part		no name	FractionalPart[x]
	integer part		no name	IntegerPart[x]
	nearest integer function	--	--	Round[x]
	quotient	--	--	Quotient[m, n]

The (possibly scaled) periodic waveform corresponding to the latter definition is known as the sawtooth wave.

The fractional part of , illustrated above, has the interesting analytic integrals

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)

The integral

(10)

is therefore a telescoping sum given by

			(11)
			(12)
			(13)

where is the Euler-Mascheroni constant and is the harmonic number.

An additional related integral that can be done in closed form and gives the same result is

(14)

(Havil 2003, pp. 109-111).

The plot above shows the fractional parts of for , showing characteristic gaps (Trott 2004, p. 223).

For the floor-based definition , a consequence of Weyl's criterion is that the sequence is dense and equidistributed in the interval for irrational , where , 2, ... (Finch 2003).