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Ceiling Function


CeilingFunction

The function [x] which gives the smallest integer >=x, 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 et al. 1994).

CeilingReImAbs
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The ceiling function is implemented in the Wolfram Language as Ceiling[z], where it is generalized to complex values of z as illustrated above.

Although some authors used the symbol ]x[ to denote the ceiling function (by analogy with the older notation [x] for the floor function), this practice is strongly discouraged (Graham et al. 1994, p. 67). Also strongly discouraged is the use of the symbol {x} 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 the fractional part of x.

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).

notationnameS&OGraham et al. Wolfram Language
[x]ceiling function--ceiling, least integerCeiling[x]
mod(m,n)congruence----Mod[m, n]
|_x_|floor functionInt(x)floor, greatest integer, integer partFloor[x]
x-|_x_|fractional valuefrac(x)fractional part or {x}SawtoothWave[x]
sgn(x)(|x|-|_|x|_|)fractional partFp(x)no nameFractionalPart[x]
sgn(x)|_|x|_|integer partIp(x)no nameIntegerPart[x]
nint(x)nearest integer function----Round[x]
m\nquotient----Quotient[m, n]

See also

Floor Function, Fractional Part, Integer Part, Mills' Constant, Mod, Nearest Integer Function, Power Ceilings, Quotient, Staircase Function

Related Wolfram sites

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

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References

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.Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994.Iverson, K. E. A Programming Language. New York: Wiley, p. 12, 1962.Schroeder, M. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, p. 57, 1991.Spanier, J.; Myland, J.; and Oldham, K. B. An Atlas of Functions, 2nd ed. Washington, DC: Hemisphere, 1987.

Referenced on Wolfram|Alpha

Ceiling Function

Cite this as:

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

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