Floor Function

The floor function , also called the greatest integer function or integer value (Spanier and Oldham 1987), gives the largest integer less than or equal to . The name and symbol for the floor function were coined by K. E. Iverson (Graham et al. 1994).

Unfortunately, in many older and current works (e.g., Honsberger 1976, p. 30; Steinhaus 1999, p. 300; Shanks 1993; Ribenboim 1996; Hilbert and Cohn-Vossen 1999, p. 38; Hardy 1999, p. 18), the symbol is used instead of (Graham et al. 1994, p. 67). In fact, this notation harks back to Gauss in his third proof of quadratic reciprocity in 1808. However, because of the elegant symmetry of the floor function and ceiling function symbols and , and because is such a useful symbol when interpreted as an Iverson bracket, the use of to denote the floor function should be deprecated. In this work, the symbol is used to denote the nearest integer function since it naturally falls between the and symbols.

Min Max Re Im

The floor function is implemented in the Wolfram Language as Floor[z], where it is generalized to complex values of 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 floor function satisfies the identity

(1)

for all integers .

A number of geometric-like sequences with a floor function in the numerator can be done analytically. For instance, sums of the form

(2)

can be done analytically for rational . For a unit fraction,

(3)

Sums of this form lead to Devil's staircase-like behavior.

For irrational , continued fraction convergents , and ,

(4)

(Borwein et al. 2004, p. 12). This leads to the rather amazing result relating sums of the floor function of multiples of to the continued fraction of by

(5)

(Mahler 1929; Borwein et al. 2004, p. 12).