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
Sums of this form lead toDevils staircase-like behavior.
Iverson, K.E.A Programming Language.New York: Wiley, p.12, 1962.
2004, p.12). This leads to the rather amazing result relating sums of the floor function of multiples of
Steinhaus, H.Mathematical Snapshots, 3rd ed.New York: Dover, 1999.
to denote the floor function should be deprecated. In this work, the symbol
Croft, H.T.; Falconer, K.J.; and Guy, R.K.Unsolved Problems in Geometry.New York: Springer-Verlag, p.2, 1991.
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Hilbert, D. and Cohn-Vossen, S.Geometry and the Imagination.New York: Chelsea, 1999.
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The floor function is implemented in theWolfram Languageas], where it is generalized to complex values of
. The name and symbol for the floor function were coined by K.E.Iverson (Graham
Shanks, D.Solved and Unsolved Problems in Number Theory, 4th ed.New York: Chelsea, p.14, 1993.
A number of geometric-like sequences with a floor function in the numerator can be done analytically. For instance, sums of the form
Borwein, J.; Bailey, D.; and Girgensohn, R.Experimentation in Mathematics: Computational Paths to Discovery.Wellesley, MA: A K Peters, 2004.
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 andceiling functionsymbols
Spanier, J. and Oldham, K.B. The Integer-Value Int(
Weisstein, Eric W.Floor Function. FromMathWorld–A Wolfram Web Resource.
Graham, R.L.; Knuth, D.E.; and Patashnik, O. Integer Functions. Ch.3 inConcrete Mathematics: A Foundation for Computer Science, 2nd ed.Reading, MA: Addison-Wesley, pp.67-101, 1994.
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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).
is used to denote thenearest integer functionsince it naturally falls between the
can be done analytically for rational
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Washington, DC: Hemisphere, pp.71-78, 1987.
Ceiling FunctionDevils StaircaseFractional PartInteger PartIverson BracketModNearest Integer FunctionPower FloorsQuotientShift TransformationStaircase Function
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Ribenboim, P.The New Book of Prime Number Records.New York: Springer-Verlag, pp.180-182, 1996.
The floor function satisfies the identity
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Honsberger, R.Mathematical Gems II.Washington, DC: Math. Assoc. Amer., 1976.
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(Mahler 1929; Borweinet al.2004, p.12).
is such a useful symbol when interpreted as anIverson bracket, the use of
, also called the greatest integer function or integer value (Spanier and Oldham 1987), gives the largestintegerless than or equal to
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Mahler, K. Arithmetische Eigenschaften der Lsungen einer Klasse von Funktionalgleichungen.Math. Ann.101, 342-366, 1929.
Hardy, G.H.Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed.New York: Chelsea, 1999.
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