Ceiling Function

for thefloor function), this practice is strongly discouraged (Graham1994, p.67). Also strongly discouraged is the use of the symbol

to denote the ceiling function (e.g., Harary 1994, pp.91, 93, and 118-119), since this same symbol is more commonly used to denote thefractional partof

Floor FunctionFractional PartInteger PartMills ConstantModNearest Integer FunctionPower CeilingsQuotientStaircase Function

, shown as the thick curve in the above plot. Schroeder (1991) calls the ceiling function symbols the gallows because of the similarity in appearance to the structure used for hangings. The name and symbol for the ceiling function were coined by K.E.Iverson (Graham

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floor, greatest integer, integer part

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).

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Iverson, K.E.A Programming Language.New York: Wiley, p.12, 1962.

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Weisstein, Eric W.Ceiling Function. FromMathWorld–A Wolfram Web Resource.

Although some authors used the symbol

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Harary, F.Graph Theory.Reading, MA: Addison-Wesley, 1994.

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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Spanier, J.; Myland, J.; and Oldham, K.B.An Atlas of Functions, 2nd ed.Washington, DC: Hemisphere, 1987.

Schroeder, M.Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise.New York: W.H. Freeman, p.57, 1991.

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Croft, H.T.; Falconer, K.J.; and Guy, R.K.Unsolved Problems in Geometry.New York: Springer-Verlag, p.2, 1991.

to denote the ceiling function (by analogy with the older notation

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The ceiling function is implemented in theWolfram Languageas], where it is generalized to complex values of

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