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Floor and ceiling functions 1
Floor and ceiling functions
 Floor function  Ceiling functionIn mathematics and computer science, the floor and ceiling functions map a real number to the largest previous orthe smallest following integer, respectively. More precisely, floor(x) = is the largest integer not greater than xand ceiling(x) = is the smallest integer not less than x.[1]
NotationGauss introduced the square bracket notation [x] for the floor function in his third proof of quadratic reciprocity(1808).[2] This remained the standard[3] in mathematics until Iverson introduced the names "floor" and "ceiling" andthe corresponding notations x and x in his 1962 book A Programming Language.[4] [5] Both notations arenow used in mathematics; this article follows Iverson.[6]
The floor function is also called the greatest integer or entier (French for "integer") function, and its value at x iscalled the integral part or integer part of x; for negative values of x the latter terms are sometimes instead taken tobe the value of the ceiling function, i.e., the value of x rounded to an integer towards 0. The language APL(programming language) uses ⌊x; other computer languages commonly use notations like entier(x) (Algol), INT(x)(BASIC), or floor(x)(C and C++).[7] In mathematics, it can also be written with boldface or double brackets .[8]
The ceiling function is usually denoted by ceil(x) or ceiling(x) in nonAPL computer languages that have a notationfor this function. In mathematics, there is another notation with reversed boldface or double brackets or justusing normal reversed brackets ]x[.[9]
The fractional part sawtooth function, denoted by for real x, is defined by the formula[10]
For all x,
Examples
Floor and ceiling functions 2
Sample value Floor Ceiling Fractional part
12/5 = 2.4 2 3 2/5 = 0.4
2.7 2 3 0.7
−2.7 −3 −2 0.3
−2 −2 −2 0
See below for the definition of the fractional part function.
Definition and propertiesIn the following formulas, x and y are real numbers, k, m, and n are integers, and is the set of integers (positive,negative, and zero).Floor and ceiling may be defined by the set equations
Since there is exactly one integer in a halfopen interval of length one, for any real x there are unique integers m andn satisfying
Then and may also be taken as the definition of floor and ceiling.
EquivalencesThese formulas can be used to simplify expressions involving floors and ceilings.[11]
In the language of order theory, the floor function is a residuated mapping, that is, part of a Galois connection: it isthe upper adjoint of the function that embeds the integers into the reals.
These formulas show how adding integers to the arguments affect the functions:
The above are not necessarily true if n is not an integer; however:
Floor and ceiling functions 3
Relations among the functionsIt is clear from the definitions that
with equality if and only if x is an integer, i.e.
In fact, since for integers n:
Negating the argument switches floor and ceiling and changes the sign:
i.e.
Negating the argument complements the fractional part:
The floor, ceiling, and fractional part functions are idempotent:
The result of nested floor or ceiling functions is the innermost function:
For fixed y, x mod y is idempotent:
Also, from the definitions,
QuotientsIf m and n are integers and n ≠ 0,
If n is positive[12]
Floor and ceiling functions 4
If m is positive[13]
For m = 2 these imply
More generally,[14] for positive m
The following can be used to convert floors to ceilings and viceversa (m positive)[15]
If m and n are positive and coprime, then
Since the righthand side is symmetrical in m and n, this implies that
More generally, if m and n are positive,
This is sometimes called a reciprocity law.[16]
Nested divisionsFor positive integers m,n, and arbitrary real number x:
Floor and ceiling functions 5
Continuity
None of the functions discussed in this article are continuous, but all are piecewise linear. and arepiecewise constant functions, with discontinuites at the integers. also has discontinuites at the integers, and
as a function of x for fixed y is discontinuous at multiples of y. is upper semicontinuous and and are lower semicontinuous. x mod y is lower
semicontinuous for positive y and upper semicontinuous for negative y.
Series expansionsSince none of the functions discussed in this article are continuous, none of them have a power series expansion.Since floor and ceiling are not periodic, they do not have Fourier series expansions.x mod y for fixed y has the Fourier series expansion[17]
in particular {x} = x mod 1 is given by
At points of discontinuity, a Fourier series converges to a value that is the average of its limits on the left and theright, unlike the floor, ceiling and fractional part functions: for y fixed and x a multiple of y the Fourier series givenconverges to y/2, rather than to x mod y = 0. At points of continuity the series converges to the true value.Using the formula {x} = x − floor(x), floor(x) = x − {x} gives
Applications
mod operatorThe mod operator, denoted by x mod y for real x and y, y ≠ 0, is defined by the formula
x mod y is always between 0 and y; i.e.if y is positive,
and if y is negative,
If x is an integer and y is a positive integer,
x mod y for a fixed y is a sawtooth function.
Floor and ceiling functions 6
Quadratic reciprocityGauss's third proof of quadratic reciprocity, as modified by Eisenstein, has two basic steps.[18] [19]
Let p and q be distinct positive odd prime numbers, and let
First, Gauss's lemma is used to show that the Legendre symbols are given by
and
The second step is to use a geometric argument to show that
Combining these formulas gives quadratic reciprocity in the form
There are formulas that use floor to express the quadratic character of small numbers mod odd primes p:[20]
Rounding
The ordinary rounding of the positive number x to the nearest integer can be expressed as The ordinaryrounding of the negative number x to the nearest integer can be expressed as
Truncation
The truncation of a nonnegative number is given by The truncation of a nonpositive number is given by .The truncation of any real number can be given by: , where sgn(x) is the sign function.
Number of digitsThe number of digits in base b of a positive integer k is
Floor and ceiling functions 7
Factors of factorialsLet n be a positive integer and p a positive prime number. The exponent of the highest power of p that divides n! isgiven by the formula[21]
Note that this is a finite sum, since the floors are zero when pk > n.
Beatty sequenceThe Beatty sequence shows how every positive irrational number gives rise to a partition of the natural numbers intotwo sequences via the floor function.[22]
Euler's constant (γ)There are formulas for Euler's constant γ = 0.57721 56649 ... that involve the floor and ceiling, e.g.[23]
and
Riemann function (ζ)The fractional part function also shows up in integral representations of the Riemann zeta function. It isstraightforward to prove (using integration by parts)[24] that if φ(x) is any function with a continuous derivative inthe closed interval [a, b],
Letting φ(n) = n−s for real part of s greater than 1 and letting a and b be integers, and letting b approach infinity gives
This formula is valid for all s with real part greater than −1, (except s = 1, where there is a pole) and combined withthe Fourier expansion for {x} can be used to extend the zeta function to the entire complex plane and to prove itsfunctional equation.[25]
For s = σ + i t in the critical strip (i.e. 0 < σ < 1),
In 1947 van der Pol used this representation to construct an analogue computer for finding roots of the zetafunction.[26]
Floor and ceiling functions 8
Formulas for prime numbersn is a prime if and only if[27]
Let r > 1 be an integer, pn be the nth prime, and define
Then[28]
There is a number θ = 1.3064... (Mill's constant) with the property that
are all prime.[29]
There is also a number ω = 1.9287800... with the property that
are all prime.[29]
π(x) is the number of primes less than or equal to x. It is a straightforward deduction from Wilson's theorem that[30]
Also, if n ≥ 2,[31]
None of the formulas in this section is of any practical use.
Solved problemRamanujan submitted this problem to the Journal of the Indian Mathematical Society.[32]
If n is a positive integer, prove that
(i)
(ii)
(iii)
Floor and ceiling functions 9
Unsolved problemThe study of Waring's problem has led to an unsolved problem:Are there any positive integers k, k ≥ 6, such that[33]
Mahler[34] has proved there can only be a finite number of such k; none are known.
Computer implementations
C and C++
Int function from floatingpoint conversion
The C, C++, and related programming languages have standardfunctions floor() [35] and ceil().[36] In these languages, conversion offloating point values to integers using type casting syntax truncatestoward zero.[37]
Python
Python follows the standard mathematical definition with integerdivision and its math.floor and ceil function, giving a result headingtowards negative infinity in all cases.[38] Python 3.0 doesn't haveinteger division, so the floor and ceil functions would have to be usedwith that version.[39]
Spreadsheet software
Most spreadsheet programs support some form of a ceiling function. Although the details differ between programs,most implementations support a second parameter—a multiple of which the given number is to be rounded to. As atypical example, ceiling(2, 3) would round 2 up to the nearest multiple of 3, so this would return 3. The definition ofwhat "round up" means, however, differs from program to program.
Microsoft Excel's ceiling function does not follow the mathematical definition, but rather as with (int) operator in C,it is a mixture of the floor and ceiling function: for x ≥ 0 it returns floor(x), and for x < 0 it returns ceiling(x). Thishas followed through to the Office Open XML file format. For example, CEILING(4.5) returns −5. A mathematicalceiling function can be emulated in Excel by using the formula "INT(value)" (please note that this is not a generalrule, as it depends on Excel's INT function, which behaves differently than most programming languages).The OpenDocument file format, as used by OpenOffice.org and others, follows the mathematical definition ofceiling for its ceiling function, with an optional parameter for Excel compatibility. For example, CEILING(4.5)returns −4.
Floor and ceiling functions 10
TypesettingThe floor and ceiling function are usually typeset with left and right square brackets where the upper (for floorfunction) or lower (for ceiling function) horizontal bars are missing, and, e.g., in the LaTeX typesetting system thesesymbols can be specified with the \lfloor, \rfloor, \lceil and \rceil commands in math mode. HTML 4.0 uses the samenames: ⌊, ⌋, ⌈, and ⌉. Unicode contains codepoints for these symbols at U+2308–U+230B:⌈x⌉, ⌊x⌋.
Notes[1] Graham, Knuth, & Patashnik, Ch. 3.1[2] Lemmermeyer, pp. 10, 23.[3] e.g. Cassels, Hardy & Wright, and Ribenboim use Gauss's notation, Graham, Knuth & Patashnik, and Crandall & Pomerance use Iverson's.[4] Iverson, p. 12.[5] Higham, p. 25.[6] See the Wolfram MathWorld article.[7] Sullivan, p. 86.[8] Mathwords: Floor Function (http:/ / www. mathwords. com/ f/ floor_function. htm).[9] Mathwords: Ceiling Function (http:/ / www. mathwords. com/ c/ ceiling_function. htm)[10] Graham, Knuth, & Patashnik, p. 70.[11] Graham, Knuth, & Patashink, Ch. 3[12] Graham, Knuth, & Patashnik, p. 72[13] Graham, Knuth, & Patashnik, p. 85[14] Graham, Knuth, & Patashnik, p. 85 and Ex. 3.15[15] Graham, Knuth, & Patashnik, Ex. 3.12[16] Graham, Knuth, & Patashnik, p. 94[17] Titchmarsh, p. 15, Eq. 2.1.7[18] Lemmermeyer, § 1.4, Ex. 1.32–1.33[19] Hardy & Wright, §§ 6.11–6.13[20] Lemmermeyer, p. 25[21] Hardy & Wright, Th. 416[22] Graham, Knuth, & Patashnik, pp. 77–78[23] These formulas are from the Wikipedia article Euler's constant, which has many more.[24] Titchmarsh, p. 13[25] Titchmarsh, pp.14–15[26] Crandall & Pomerance, p. 391[27] Crandall & Pomerance, Ex. 1.3, p. 46[28] Hardy & Wright, § 22.3[29] Ribenboim, p. 186[30] Ribenboim, p. 181[31] Crandall & Pomerance, Ex. 1.4, p. 46[32] Ramanujan, Question 723, Papers p. 332[33] Hardy & Wright, p. 337[34] Mahler, K. On the fractional parts of the powers of a rational number II, 1957, Mathematika, 4, pages 122124[35] "C++ reference of floor function" (http:/ / cppreference. com/ wiki/ numeric/ c/ floor). . Retrieved 5 December 2010.[36] "C++ reference of ceil function" (http:/ / cppreference. com/ wiki/ numeric/ c/ ceil). . Retrieved 5 December 2010.[37] ISO standard for C, § 6.3.1.4, p. 43.[38] http:/ / docs. python. org/ library/ math. html?highlight=ceil#math. ceil[39] http:/ / www. python. org/ dev/ peps/ pep0238/
Floor and ceiling functions 11
References• J.W.S. Cassels (1957), An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and
Mathematical Physics, 45, Cambridge University Press• Crandall, Richard; Pomeramce, Carl (2001), Prime Numbers: A Computational Perspective, New York: Springer,
ISBN 0387047779• Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994), Concrete Mathematics, Reading Ma.:
AddisonWesley, ISBN 0201558025• Hardy, G. H.; Wright, E. M. (1980), An Introduction to the Theory of Numbers (Fifth edition), Oxford: Oxford
University Press, ISBN 9780198531715• Nicholas J. Higham, Handbook of writing for the mathematical sciences, SIAM. ISBN 0898714206, p. 25• ISO/IEC. ISO/IEC 9899::1999(E): Programming languages — C (2nd ed), 1999; Section 6.3.1.4, p. 43.• Iverson, Kenneth E. (1962), A Programming Language, Wiley• Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer, ISBN 3540669674• Ramanujan, Srinivasa (2000), Collected Papers, Providence RI: AMS / Chelsea, ISBN 9780821820766• Ribenboim, Paulo (1996), The New Book of Prime Number Records, New York: Springer, ISBN 0387944575• Michael Sullivan. Precalculus, 8th edition, p. 86• Titchmarsh, Edward Charles; HeathBrown, David Rodney ("Roger") (1986), The Theory of the Riemann
Zetafunction (2nd ed.), Oxford: Oxford U. P., ISBN 0198533691
External links• Štefan Porubský, "Integer rounding functions" (http:/ / www. cs. cas. cz/ portal/ AlgoMath/ NumberTheory/
ArithmeticFunctions/ IntegerRoundingFunctions. htm), Interactive Information Portal for AlgorithmicMathematics, Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic,retrieved 24 October 2008
• Weisstein, Eric W., " Floor Function (http:/ / mathworld. wolfram. com/ FloorFunction. html)" from MathWorld.• Weisstein, Eric W., " Ceiling Function (http:/ / mathworld. wolfram. com/ CeilingFunction. html)" from
MathWorld.
Article Sources and Contributors 12
Article Sources and ContributorsFloor and ceiling functions Source: http://en.wikipedia.org/w/index.php?oldid=408024624 Contributors: 08glasgow08, 0rrAvenger, 1exec1, ABF, Aaron Rotenberg, Abdull, AlexanderIwaschkin, Anonymous Dissident, Archelon, Arvindn, AxelBoldt, Beland, BenFrantzDale, BiT, CanadianLinuxUser, Charles Matthews, Chris857, Clark Kimberling, DRE, Da Joe, Davzier,Dbenbenn, DePiep, Dhollm, Duja, Dysprosia, EdC, Eirik, El C, Elassint, Enochlau, FStHilaire, Faradayplank, Firespade, Fnielsen, Fredrik, FvdP, Gavia immer, Gesslein, Giftlite, Henrygb,Hvn0413, Ideyal, Jkl, Johnflux, Jstaniek, Julendra, Keka, Kieff, KnowledgeOfSelf, KyleP, LOL, Lambiam, Lester, LilHelpa, Linas, Looxix, Macrakis, Maksime, Maltelauridsbrigge, Mandarax,Marc van Leeuwen, Martnym, Mat cross, Mav, McKay, Mgccl, Michael Hardy, Mohamed AlDabbagh, Mrand, Murtasa, Mwtoews, Myblackmouse, Nk, Ocolon, Octahedron80, OlegAlexandrov, Oliphaunt, Omegatron, Paepaok, Patrick, Peterl, Piano non troppo, Quantling, RDBury, Rbj, Reg Holmes, RexNL, Reyk, Rich Farmbrough, Rifleman 82, Robprain, Sabbut, Sagie,Schutz, Shagriz, Skorpionek, Smalljim, Snoyes, Spoon!, Super Rad!, SvNH, Takwan, The Thing That Should Not Be, Thincat, Timmyp320, Tinctorius, Tjdw, Tosha, Usien6, VKokielov,Vanish2, Verin Sedai, VirginiaAmerican, WDavis1911, Weihao.chiu, Wich, Wimmerm, Wisling, Wj32, XJamRastafire, Zrs 12, 146 anonymous edits
Image Sources, Licenses and ContributorsImage:Floor function.svg Source: http://en.wikipedia.org/w/index.php?title=File:Floor_function.svg License: unknown Contributors: Darapti, Omegatron, Pieter KuiperImage:Ceiling function.svg Source: http://en.wikipedia.org/w/index.php?title=File:Ceiling_function.svg License: unknown Contributors: Cwbm (commons), Darapti, Omegatron, PieterKuiper, 1 anonymous editsFile:Int function.svg Source: http://en.wikipedia.org/w/index.php?title=File:Int_function.svg License: unknown Contributors: Darapti, Omegatron
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