Fractional Part

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).

Min   Max       Re       Im

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).

A consequence of Weyl's criterion is that the sequence  is dense and equidistributed in the interval  for irrational , where , 2, ... (Finch 2003).

REFERENCES:

Finch, S. R. "Powers of 3/2 Modulo One." §2.30.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 194-199, 2003.

Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 109-110, 2003.

Miklavc, A. "Elementary Proofs of Two Theorems on the Distribution of Numbers  (mod 1)." Proc. Amer. Math. Soc. 39, 279-280, 1973.

Sloane, N. J. A. Sequence A000079/M1129 in "The On-Line Encyclopedia of Integer Sequences."

Spanier, J. and Oldham, K. B. "The Integer-Value Int() and Fractional-Value frac() Functions." Ch. 9 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 71-78, 1987.

Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. https://www.mathematicaguidebooks.org/.