Ceiling Function
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).
![👁 [x]](https://mathworld.wolfram.com/images/equations/CeilingFunction/Inline1.svg){kind=link}
The ceiling function is implemented in the Wolfram Language as [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
.
![👁 ]x[](https://mathworld.wolfram.com/images/equations/CeilingFunction/Inline4.svg){kind=link}
![👁 [x]](https://mathworld.wolfram.com/images/equations/CeilingFunction/Inline5.svg){kind=link}
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 |
| 👁 [x] | ceiling function | -- | ceiling, least integer | [x] |
| 👁 mod(m,n) | congruence | -- | -- | [m, n] |
| 👁 |_x_| | floor function | 👁 Int(x) | floor, greatest integer, integer part | [x] |
| 👁 x-|_x_| | fractional value | 👁 frac(x) | fractional
part or 👁 {x} | [x] |
| 👁 sgn(x)(|x|-|_|x|_|) | fractional part | 👁 Fp(x) | no name | [x] |
| 👁 sgn(x)|_|x|_| | integer part | 👁 Ip(x) | no name | [x] |
| 👁 nint(x) | nearest integer function | -- | -- | [x] |
| 👁 m\n | quotient | -- | -- | [m, n] |
![👁 [x]](https://mathworld.wolfram.com/images/equations/CeilingFunction/Inline8.svg){kind=link}
See also
Floor Function, Fractional Part, Integer Part, Mills' Constant, Mod, Nearest Integer Function, Power Ceilings, Quotient, Staircase FunctionRelated Wolfram sites
http://functions.wolfram.com/IntegerFunctions/Ceiling/Explore with Wolfram|Alpha
More things to try:
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 FunctionCite this as:
Weisstein, Eric W. "Ceiling Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CeilingFunction.html