Floor and ceiling functions
Inmathematics,thefloor functionis thefunctionthat takes as input areal numberx,and gives as output the greatestintegerless than or equal tox,denoted⌊x⌋orfloor(x).Similarly, theceiling functionmapsxto the smallest integer greater than or equal tox,denoted⌈x⌉orceil(x).[1]
For example, for floor:⌊2.4⌋ = 2,⌊−2.4⌋ = −3,and for ceiling:⌈2.4⌉ = 3,and⌈−2.4⌉ = −2.
The floor ofxis also called theintegral part,integer part,greatest integer,orentierofx,and was historically denoted[x](among other notations).[2]However, the same term,integer part,is also used fortruncationtowards zero, which differs from the floor function for negative numbers.
Fornan integer,⌊n⌋ = ⌈n⌉ =n.
Althoughfloor(x+1)andceil(x)produce graphs that appear exactly alike, they are not the same when the value of x is an exact integer. For example, whenx=2.0001;⌊2.0001+1⌋ = ⌈2.0001⌉ = 3.However, ifx=2, then⌊2+1⌋ = 3,while⌈2⌉ = 2.
x | Floor⌊x⌋ | Ceiling⌈x⌉ | Fractional part{x} |
---|---|---|---|
2 | 2 | 2 | 0 |
2.0001 | 2 | 3 | 0.0001 |
2.4 | 2 | 3 | 0.4 |
2.9 | 2 | 3 | 0.9 |
2.999 | 2 | 3 | 0.999 |
−2.7 | −3 | −2 | 0.3 |
−2 | −2 | −2 | 0 |
Notation
[edit]Theintegral partorinteger partof a number (partie entièrein the original) was first defined in 1798 byAdrien-Marie Legendrein his proof of theLegendre's formula.
Carl Friedrich Gaussintroduced the square bracket notation[x]in his third proof ofquadratic reciprocity(1808).[3]This remained the standard[4]in mathematics untilKenneth E. Iversonintroduced, in his 1962 bookA Programming Language,the names "floor" and "ceiling" and the corresponding notations⌊x⌋and⌈x⌉.[5][6](Iverson used square brackets for a different purpose, theIverson bracketnotation.) Both notations are now used in mathematics, although Iverson's notation will be followed in this article.
In some sources, boldface or double brackets⟦x⟧are used for floor, and reversed brackets⟧x⟦or]x[for ceiling.[7][8]
Thefractional partis thesawtooth function,denoted by{x}for realxand defined by the formula
- {x} =x− ⌊x⌋[9]
For allx,
- 0 ≤ {x} < 1.
These characters are provided in Unicode:
- U+2308⌈LEFT CEILING(⌈, ⌈)
- U+2309⌉RIGHT CEILING(⌉, ⌉)
- U+230A⌊LEFT FLOOR(⌊, ⌊)
- U+230B⌋RIGHT FLOOR(⌋, ⌋)
In theLaTeXtypesetting system, these symbols can be specified with the\lceil, \rceil, \lfloor,
and\rfloor
commands in math mode. LaTeX has supported UTF-8 since 2018, so the Unicode characters can now be used directly.[10]Larger versions are\left\lceil, \right\rceil, \left\lfloor,
and\right\rfloor
.
Definition and properties
[edit]Given real numbersxandy,integersmandnand the set ofintegers,floor and ceiling may be defined by the equations
Since there is exactly one integer in ahalf-open intervalof length one, for any real numberx,there are unique integersmandnsatisfying the equation
whereandmay also be taken as the definition of floor and ceiling.
Equivalences
[edit]These formulas can be used to simplify expressions involving floors and ceilings.[11]
In the language oforder theory,the floor function is aresiduated mapping,that is, part of aGalois connection:it is the upper adjoint of the function that embeds the integers into the reals.
These formulas show how adding an integernto the arguments affects the functions:
The above are never true ifnis not an integer; however, for everyxandy,the following inequalities hold:
Monotonicity
[edit]Both floor and ceiling functions aremonotonically non-decreasing functions:
Relations among the functions
[edit]It is clear from the definitions that
- with equality if and only ifxis an integer, i.e.
In fact, for integersn,both floor and ceiling functions are theidentity:
Negating the argument switches floor and ceiling and changes the sign:
and:
Negating the argument complements the fractional part:
The floor, ceiling, and fractional part functions areidempotent:
The result of nested floor or ceiling functions is the innermost function:
due to the identity property for integers.
Quotients
[edit]Ifmandnare integers andn≠ 0,
Ifnis a positive integer[12]
Ifmis positive[13]
Form= 2 these imply
More generally,[14]for positivem(SeeHermite's identity)
The following can be used to convert floors to ceilings and vice versa (mpositive)[15]
For allmandnstrictly positive integers:[16]
which, for positive andcoprimemandn,reduces to
and similarly for the ceiling and fractional part functions (still for positive andcoprimemandn),
Since the right-hand side of the general case is symmetrical inmandn,this implies that
More generally, ifmandnare positive,
This is sometimes called areciprocity law.[17]
Division by positive integers gives rise to an interesting and sometimes useful property. Assuming,
Similarly,
Indeed,
keeping in mind that The second equivalence involving the ceiling function can be proved similarly.
Nested divisions
[edit]For positive integern,and arbitrary real numbersm,x:[18]
Continuity and series expansions
[edit]None of the functions discussed in this article arecontinuous,but all arepiecewise linear:the functions,,andhave discontinuities at the integers.
isupper semi-continuousandandare lower semi-continuous.
Since none of the functions discussed in this article are continuous, none of them have apower seriesexpansion. Since floor and ceiling are not periodic, they do not have uniformly convergentFourier seriesexpansions. The fractional part function has Fourier series expansion[19] forxnot an integer.
At points of discontinuity, a Fourier series converges to a value that is the average of its limits on the left and the right, unlike the floor, ceiling and fractional part functions: foryfixed andxa multiple ofythe Fourier series given converges toy/2, rather than toxmody= 0. At points of continuity the series converges to the true value.
Using the formulagives forxnot an integer.
Applications
[edit]Mod operator
[edit]For an integerxand a positive integery,themodulo operation,denoted byxmody,gives the value of the remainder whenxis divided byy.This definition can be extended to realxandy,y≠ 0, by the formula
Then it follows from the definition of floor function that this extended operation satisfies many natural properties. Notably,xmodyis always between 0 andy,i.e.,
ifyis positive,
and ifyis negative,
Quadratic reciprocity
[edit]Gauss's third proof ofquadratic reciprocity,as modified by Eisenstein, has two basic steps.[20][21]
Letpandqbe distinct positive odd prime numbers, and let
First,Gauss's lemmais used to show that theLegendre symbolsare given by
The second step is to use ageometricargument 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 primesp:[22]
Rounding
[edit]For an arbitrary real number,roundingto the nearest integer withtie breakingtowards positive infinity is given by;rounding towards negative infinity is given as.
If tie-breaking is away from 0, then the rounding function is(seesign function), androunding towards evencan be expressed with the more cumbersome,which is the above expression for rounding towards positive infinityminus anintegralityindicatorfor.
Rounding areal numberto the nearest integer value forms a very basic type ofquantizer– auniformone. A typical (mid-tread) uniform quantizer with a quantizationstep sizeequal to some valuecan be expressed as
- ,
Number of digits
[edit]The number of digits inbasebof a positive integerkis
Number of strings without repeated characters
[edit]The number of possiblestringsof arbitrary length that doesn't use any character twice is given by[23][better source needed]
where:
- n> 0 is the number of letters in the Alpha bet (e.g., 26 inEnglish)
- thefalling factorialdenotes the number of strings of lengthkthat don't use any character twice.
- n!denotes thefactorialofn
- e= 2.718... isEuler's number
Forn= 26, this comes out to 1096259850353149530222034277.
Factors of factorials
[edit]Letnbe a positive integer andpa positive prime number. The exponent of the highest power ofpthat dividesn!is given by a version ofLegendre's formula[24]
whereis the way of writingnin basep.This is a finite sum, since the floors are zero whenpk>n.
Beatty sequence
[edit]TheBeatty sequenceshows how every positiveirrational numbergives rise to a partition of thenatural numbersinto two sequences via the floor function.[25]
Euler's constant (γ)
[edit]There are formulas forEuler's constantγ = 0.57721 56649... that involve the floor and ceiling, e.g.[26]
and
Riemann zeta function (ζ)
[edit]The fractional part function also shows up in integral representations of theRiemann zeta function.It is straightforward to prove (using integration by parts)[27]that ifis any function with a continuous derivative in the closed interval [a,b],
Lettingforreal partofsgreater than 1 and lettingaandbbe integers, and lettingbapproach infinity gives
This formula is valid for allswith real part greater than −1, (excepts= 1, where there is a pole) and combined with the Fourier expansion for {x} can be used to extend the zeta function to the entire complex plane and to prove its functional equation.[28]
Fors=σ+itin the critical strip 0 <σ< 1,
In 1947van der Polused this representation to construct an analogue computer for finding roots of the zeta function.[29]
Formulas for prime numbers
[edit]The floor function appears in several formulas characterizing prime numbers. For example, sinceis equal to 1 ifmdividesn,and to 0 otherwise, it follows that a positive integernis a primeif and only if[30]
One may also give formulas for producing the prime numbers. For example, letpnbe then-th prime, and for any integerr> 1, define the real numberαby the sum
Then[31]
A similar result is that there is a numberθ= 1.3064... (Mills' constant) with the property that
are all prime.[32]
There is also a numberω= 1.9287800... with the property that
are all prime.[32]
Letπ(x) be the number of primes less than or equal tox.It is a straightforward deduction fromWilson's theoremthat[33]
Also, ifn≥ 2,[34]
None of the formulas in this section are of any practical use.[35][36]
Solved problems
[edit]Ramanujansubmitted these problems to theJournal of the Indian Mathematical Society.[37]
Ifnis a positive integer, prove that
Some generalizations to the above floor function identities have been proven.[38]
Unsolved problem
[edit]The study ofWaring's problemhas led to an unsolved problem:
Are there any positive integersk≥ 6 such that[39]
Mahlerhas proved there can only be a finite number of suchk;none are known.[40]
Computer implementations
[edit]In most programming languages, the simplest method to convert a floating point number to an integer does not do floor or ceiling, but truncation. The reason for this is historical, as the first machines usedones' complementand truncation was simpler to implement (floor is simpler intwo's complement).FORTRANwas defined to require this behavior and thus almost all processors implement conversion this way. Some consider this to be an unfortunate historical design decision that has led to bugs handling negative offsets and graphics on the negative side of the origin.[citation needed]
Anarithmetic right-shiftof a signed integerbyis the same as.Division by a power of 2 is often written as a right-shift, not for optimization as might be assumed, but because the floor of negative results is required. Assuming such shifts are "premature optimization" and replacing them with division can break software.[citation needed]
Many programming languages (includingC,C++,[41][42]C#,[43][44]Java,[45][46]
Julia,[47]
PHP,[48][49]R,[50]andPython[51]) provide standard functions for floor and ceiling, usually calledfloor
andceil
,or less commonlyceiling
.[52]The languageAPLuses⌊x
for floor. TheJ Programming Language,a follow-on to APL that is designed to use standard keyboard symbols, uses<.
for floor and>.
for ceiling.[53]
ALGOLusesentier
for floor.
InMicrosoft Excelthe funtionINT
rounds down rather than toward zero,[54]whileFLOOR
rounds toward zero, the opposite of what "int" and "floor" do in other languages. Since 2010FLOOR
has been changed to error if the number is negative.[55]TheOpenDocumentfile format, as used byOpenOffice.org,Libreofficeand others,INT
[56]andFLOOR
both do floor, andFLOOR
has a third argument to reproduce Excel's earlier behavior.[57]
See also
[edit]Citations
[edit]- ^Graham, Knuth, & Patashnik, Ch. 3.1
- ^
1) Luke Heaton,A Brief History of Mathematical Thought,2015,ISBN1472117158(n.p.)
2) Albert A. Blanket al.,Calculus: Differential Calculus,1968, p. 259
3) John W. Warris, Horst Stocker,Handbook of mathematics and computational science,1998,ISBN0387947469,p. 151 - ^Lemmermeyer, pp. 10, 23.
- ^e.g. Cassels, Hardy & Wright, and Ribenboim use Gauss's notation. Graham, Knuth & Patashnik, and Crandall & Pomerance use Iverson's.
- ^Iverson, p. 12.
- ^Higham, p. 25.
- ^Mathwords: Floor Function.
- ^Mathwords: Ceiling Function
- ^Graham, Knuth, & Patashnik, p. 70.
- ^"LaTeX News, Issue 28"(PDF; 379 KB).The LaTeX Project. April 2018.Retrieved27 July2024.
- ^Graham, Knuth, & Patashink, Ch. 3
- ^Graham, Knuth, & Patashnik, p. 73
- ^Graham, Knuth, & Patashnik, p. 85
- ^Graham, Knuth, & Patashnik, p. 85 and Ex. 3.15
- ^Graham, Knuth, & Patashnik, Ex. 3.12
- ^Graham, Knuth, & Patashnik, p. 94.
- ^Graham, Knuth, & Patashnik, p. 94
- ^Graham, Knuth, & Patashnik, p. 71, apply theorem 3.10 with x/m as input and the division by n as function
- ^Titchmarsh, p. 15, Eq. 2.1.7
- ^Lemmermeyer, § 1.4, Ex. 1.32–1.33
- ^Hardy & Wright, §§ 6.11–6.13
- ^Lemmermeyer, p. 25
- ^OEISsequence A000522 (Total number of arrangements of a set with n elements: a(n) = Sum_{k=0..n} n!/k!.)(See Formulas.)
- ^Hardy & Wright, Th. 416
- ^Graham, Knuth, & Patashnik, pp. 77–78
- ^These formulas are from the Wikipedia articleEuler's constant,which has many more.
- ^Titchmarsh, p. 13
- ^Titchmarsh, pp.14–15
- ^Crandall & Pomerance, p. 391
- ^Crandall & Pomerance, Ex. 1.3, p. 46. The infinite upper limit of the sum can be replaced withn.An equivalent condition isn> 1 is prime if and only if.
- ^Hardy & Wright, § 22.3
- ^abRibenboim, p. 186
- ^Ribenboim, p. 181
- ^Crandall & Pomerance, Ex. 1.4, p. 46
- ^Ribenboim, p. 180 says that "Despite the nil practical value of the formulas... [they] may have some relevance to logicians who wish to understand clearly how various parts of arithmetic may be deduced from different axiomatzations..."
- ^Hardy & Wright, pp. 344—345 "Any one of these formulas (or any similar one) would attain a different status if the exact value of the number α... could be expressed independently of the primes. There seems no likelihood of this, but it cannot be ruled out as entirely impossible."
- ^Ramanujan, Question 723,Papersp. 332
- ^Somu, Sai Teja; Kukla, Andrzej (2022)."On some generalizations to floor function identities of Ramanujan"(PDF).Integers.22.arXiv:2109.03680.
- ^Hardy & Wright, p. 337
- ^Mahler, Kurt(1957). "On the fractional parts of the powers of a rational number II".Mathematika.4(2): 122–124.doi:10.1112/S0025579300001170.
- ^"C++ reference of
floor
function ".Retrieved5 December2010. - ^"C++ reference of
ceil
function ".Retrieved5 December2010. - ^dotnet-bot."Math.Floor Method (System)".docs.microsoft.Retrieved28 November2019.
- ^dotnet-bot."Math.Ceiling Method (System)".docs.microsoft.Retrieved28 November2019.
- ^"Math (Java SE 9 & JDK 9 )".docs.oracle.Retrieved20 November2018.
- ^"Math (Java SE 9 & JDK 9 )".docs.oracle.Retrieved20 November2018.
- ^"Math (Julia v1.10)".docs.julialang.org/en/v1/.Retrieved4 September2024.
- ^"PHP manual for
ceil
function ".Retrieved18 July2013. - ^"PHP manual for
floor
function ".Retrieved18 July2013. - ^"R: Rounding of Numbers".
- ^"Python manual for
math
module ".Retrieved18 July2013. - ^Sullivan, p. 86.
- ^"Vocabulary".J Language.Retrieved6 September2011.
- ^"INT function".Retrieved29 October2021.
- ^"FLOOR function".Retrieved29 October2021.
- ^"Documentation/How Tos/Calc: INT function".Retrieved29 October2021.
- ^"Documentation/How Tos/Calc: FLOOR function".Retrieved29 October2021.
References
[edit]- J.W.S. Cassels (1957),An introduction to Diophantine approximation,Cambridge Tracts in Mathematics and Mathematical Physics, vol. 45,Cambridge University Press
- Crandall, Richard; Pomerance, Carl (2001),Prime Numbers: A Computational Perspective,New York:Springer,ISBN0-387-94777-9
- Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994),Concrete Mathematics,Reading Ma.: Addison-Wesley,ISBN0-201-55802-5
- Hardy, G. H.; Wright, E. M. (1980),An Introduction to the Theory of Numbers (Fifth edition),Oxford:Oxford University Press,ISBN978-0-19-853171-5
- Nicholas J. Higham,Handbook of writing for the mathematical sciences,SIAM.ISBN0-89871-420-6,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,ISBN3-540-66957-4
- Ramanujan, Srinivasa (2000),Collected Papers,Providence RI: AMS / Chelsea,ISBN978-0-8218-2076-6
- Ribenboim, Paulo (1996),The New Book of Prime Number Records,New York: Springer,ISBN0-387-94457-5
- Michael Sullivan.Precalculus,8th edition, p. 86
- Titchmarsh, Edward Charles; Heath-Brown, David Rodney ( "Roger" ) (1986),The Theory of the Riemann Zeta-function(2nd ed.), Oxford: Oxford U. P.,ISBN0-19-853369-1
External links
[edit]- "Floor function",Encyclopedia of Mathematics,EMS Press,2001 [1994]
- Štefan Porubský,"Integer rounding functions",Interactive Information Portal for Algorithmic Mathematics,Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic, retrieved 24 October 2008
- Weisstein, Eric W."Floor Function".MathWorld.
- Weisstein, Eric W."Ceiling Function".MathWorld.