What does the floor function do?

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.

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).

The floor function satisfies the identity

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

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

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

For irrational , continued fraction convergents , and ,

(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

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

See also

Ceiling Function, Devil's Staircase, Fractional Part, Integer Part, Iverson Bracket, Mod, Nearest Integer Function, Power Floors, Quotient, Shift Transformation, Staircase Function

Related Wolfram sites

http://functions.wolfram.com/IntegerFunctions/Floor/

Explore with Wolfram|Alpha

References

Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991.Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Integer Functions." Ch. 3 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, pp. 67-101, 1994.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, 1999.Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., 1976.Iverson, K. E. A Programming Language. New York: Wiley, p. 12, 1962.Mahler, K. "Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen." Math. Ann. 101, 342-366, 1929.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 180-182, 1996.Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 14, 1993.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.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.

Referenced on Wolfram|Alpha

Floor Function

Cite this as:

Weisstein, Eric W. "Floor Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FloorFunction.html

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