In mathematics, the floor function (or greatest integer function) is the function that takes as input a real numberx, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the smallest integer greater than or equal to x, denoted ⌈x⌉ or ceil(x).^{[1]}
Floor and ceiling functions
Floor function
Ceiling function
For example, for floor: ⌊2.4⌋ = 2, ⌊−2.4⌋ = −3, and for ceiling: ⌈2.4⌉ = 3, and ⌈−2.4⌉ = −2.
Historically, the floor of x has been–and still is–called the integral part or integer part of x, often denoted [x] (as well as a variety of other notations).^{[2]} However, the same term, integer part, is also used for truncation towards zero, which differs from the floor function for negative numbers.
For n an integer, ⌊n⌋ = ⌈n⌉ = [n] = n.
Although floor(x+1) and ceil(x) produce graphs that appear exactly alike, they are not the same when the value of x is an exact integer. For example, when x=2.0001; ⌊2.0001+1⌋ = ⌈2.0001⌉ = 3. However, if x=2, then ⌊2+1⌋ = 3, while ⌈2⌉ = 2.
The integral part or integer part of a number (partie entière in the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula.
Carl Friedrich Gauss introduced the square bracket notation [x] in his third proof of quadratic reciprocity (1808).^{[3]} This remained the standard^{[4]} in mathematics until Kenneth E. Iverson introduced, in his 1962 book A Programming Language, the names "floor" and "ceiling" and the corresponding notations ⌊x⌋ and ⌈x⌉.^{[5]}^{[6]} (Iverson used square brackets for a different purpose, the Iverson bracket notation.) 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]}
In the LaTeX typesetting system, these symbols can be specified with the \lceil, \rceil, \lfloor, and \rfloor commands in math mode, and extended in size using \left\lceil, \right\rceil, \left\lfloor, and \right\rfloor as needed.
Some authors define [x] as the round-toward-zero function^{[citation needed]}, so [2.4] = 2 and [−2.4] = −2, and call it the "integer part".
Definition and propertiesedit
Given real numbers x and y, integers m and n and the set of integers$\mathbb {Z}$, floor and ceiling may be defined by the equations
Since there is exactly one integer in a half-open interval of length one, for any real number x, there are unique integers m and n satisfying the equation
$x-1<m\leq x\leq n<x+1.$
where $\lfloor x\rfloor =m$ and $\lceil x\rceil =n$ may also be taken as the definition of floor and ceiling.
Equivalencesedit
These formulas can be used to simplify expressions involving floors and ceilings.^{[10]}
${\begin{alignedat}{3}\lfloor x\rfloor &=m\ \ &&{\mbox{ if and only if }}&m&\leq x<m+1,\\\lceil x\rceil &=n&&{\mbox{ if and only if }}&\ \ n-1&<x\leq n,\\\lfloor x\rfloor &=m&&{\mbox{ if and only if }}&x-1&<m\leq x,\\\lceil x\rceil &=n&&{\mbox{ if and only if }}&x&\leq n<x+1.\end{alignedat}}$
In the language of order theory, the floor function is a residuated mapping, that is, part of a Galois connection: it is the upper adjoint of the function that embeds the integers into the reals.
${\begin{aligned}x<n&\;\;{\mbox{ if and only if }}&\lfloor x\rfloor &<n,\\n<x&\;\;{\mbox{ if and only if }}&n&<\lceil x\rceil ,\\x\leq n&\;\;{\mbox{ if and only if }}&\lceil x\rceil &\leq n,\\n\leq x&\;\;{\mbox{ if and only if }}&n&\leq \lfloor x\rfloor .\end{aligned}}$
These formulas show how adding an integer n to the arguments affects the functions:
keeping in mind that ${\textstyle \lfloor x/n\rfloor ={\bigl \lfloor }\lfloor x\rfloor /n{\bigr \rfloor }.}$
The second equivalence involving the ceiling function can be proved similarly.
Nested divisionsedit
For positive integer n, and arbitrary real numbers m,x:^{[17]}
None of the functions discussed in this article are continuous, but all are piecewise linear: the functions $\lfloor x\rfloor$, $\lceil x\rceil$, and $\{x\}$ have discontinuities at the integers.
$\lfloor x\rfloor$ is upper semi-continuous and $\lceil x\rceil$ and $\{x\}$ are lower semi-continuous.
Since 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 uniformly convergent Fourier series expansions. The fractional part function has Fourier series expansion^{[18]}
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: for y fixed and x a multiple of y the Fourier series given converges to y/2, rather than to x mod y = 0. At points of continuity the series converges to the true value.
For an integer x and a positive integer y, the modulo operation, denoted by x mod y, gives the value of the remainder when x is divided by y. This definition can be extended to real x and y, y ≠ 0, by the formula
Then it follows from the definition of floor function that this extended operation satisfies many natural properties. Notably, x mod y is always between 0 and y, i.e.,
if y is positive,
$0\leq x{\bmod {y}}<y,$
and if y is negative,
$0\geq x{\bmod {y}}>y.$
Quadratic reciprocityedit
Gauss's third proof of quadratic reciprocity, as modified by Eisenstein, has two basic steps.^{[19]}^{[20]}
Let p and q be distinct positive odd prime numbers, and let $m={\tfrac {1}{2}}(p-1),$$n={\tfrac {1}{2}}(q-1).$
For an arbitrary real number $x$, rounding$x$ to the nearest integer with tie breaking towards positive infinity is given by ${\text{rpi}}(x)=\left\lfloor x+{\tfrac {1}{2}}\right\rfloor =\left\lceil {\tfrac {1}{2}}\lfloor 2x\rfloor \right\rceil$; rounding towards negative infinity is given as ${\text{rni}}(x)=\left\lceil x-{\tfrac {1}{2}}\right\rceil =\left\lfloor {\tfrac {1}{2}}\lceil 2x\rceil \right\rfloor$.
If tie-breaking is away from 0, then the rounding function is ${\text{ri}}(x)=\operatorname {sgn}(x)\left\lfloor |x|+{\tfrac {1}{2}}\right\rfloor$ (see sign function), and rounding towards even can be expressed with the more cumbersome $\lfloor x\rceil =\left\lfloor x+{\tfrac {1}{2}}\right\rfloor +{\bigl \lceil }{\tfrac {1}{4}}(2x-1){\bigr \rceil }-{\bigl \lfloor }{\tfrac {1}{4}}(2x-1){\bigr \rfloor }-1$, which is the above expression for rounding towards positive infinity ${\text{rpi}}(x)$ minus an integralityindicator for ${\tfrac {1}{4}}(2x-1)$.
Rounding a real number$x$ to the nearest integer value forms a very basic type of quantizer – a uniform one. A typical (mid-tread) uniform quantizer with a quantization step size equal to some value $\Delta$ can be expressed as
For n = 26, this comes out to 1096259850353149530222034277.
Factors of factorialsedit
Let n be a positive integer and p a positive prime number. The exponent of the highest power of p that divides n! is given by a version of Legendre's formula^{[23]}
The fractional part function also shows up in integral representations of the Riemann zeta function. It is straightforward to prove (using integration by parts)^{[26]} that if $\varphi (x)$ is any function with a continuous derivative in the closed interval [a, b],
This formula is valid for all s with real part greater than −1, (except s = 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.^{[27]}
In 1947 van der Pol used this representation to construct an analogue computer for finding roots of the zeta function.^{[28]}
Formulas for prime numbersedit
The floor function appears in several formulas characterizing prime numbers. For example, since ${\textstyle {\bigl \lfloor }{\frac {n}{m}}{\bigr \rfloor }-{\bigl \lfloor }{\frac {n-1}{m}}{\bigr \rfloor }}$ is equal to 1 if m divides n, and to 0 otherwise, it follows that a positive integer n is a prime if and only if^{[29]}
One may also give formulas for producing the prime numbers. For example, let p_{n} be the n-th prime, and for any integer r > 1, define the real number α by the sum
Mahler has proved there can only be a finite number of such k; none are known.^{[39]}
Computer implementationsedit
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 used ones' complement and truncation was simpler to implement (floor is simpler in two's complement). FORTRAN was 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]}
An arithmetic right-shift of a signed integer $x$ by $n$ is the same as $\left\lfloor x/2^{n}\right\rfloor$. 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 (including C, C++,^{[40]}^{[41]}C#,^{[42]}^{[43]}Java,^{[44]}^{[45]}PHP,^{[46]}^{[47]}R,^{[48]} and Python^{[49]}) provide standard functions for floor and ceiling, usually called floor and ceil, or less commonly ceiling.^{[50]} The language APL uses ⌊x for floor. The J Programming Language, a follow-on to APL that is designed to use standard keyboard symbols, uses <. for floor and >. for ceiling.^{[51]}ALGOL usesentier for floor.
In Microsoft Excel the funtion INT rounds down rather than toward zero,^{[52]} while FLOOR rounds toward zero, the opposite of what "int" and "floor" do in other languages. Since 2010 FLOOR has been changed to error if the number is negative.^{[53]} The OpenDocument file format, as used by OpenOffice.org, Libreoffice and others, INT^{[54]} and FLOOR both do floor, and FLOOR has a third argument to reproduce Excel's earlier behavior.^{[55]}
^
1) Luke Heaton, A Brief History of Mathematical Thought, 2015, ISBN 1472117158 (n.p.)
2) Albert A. Blank et al., Calculus: Differential Calculus, 1968, p. 259
3) John W. Warris, Horst Stocker, Handbook of mathematics and computational science, 1998, ISBN 0387947469, p. 151
^Crandall & Pomerance, Ex. 1.3, p. 46. The infinite upper limit of the sum can be replaced with n. An equivalent condition is n > 1 is prime if and only if ${\textstyle \sum _{m=1}^{\lfloor {\sqrt {n}}\rfloor }{\bigl (}{\bigl \lfloor }{\frac {n}{m}}{\bigr \rfloor }-{\bigl \lfloor }{\frac {n-1}{m}}{\bigr \rfloor }{\bigr )}=1}$ .
^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."
^"Documentation/How Tos/Calc: INT function". Retrieved 29 October 2021.
^"Documentation/How Tos/Calc: FLOOR function". Retrieved 29 October 2021.
Referencesedit
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, ISBN 0-387-94777-9
Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994), Concrete Mathematics, Reading Ma.: Addison-Wesley, ISBN 0-201-55802-5
Hardy, G. H.; Wright, E. M. (1980), An Introduction to the Theory of Numbers (Fifth edition), Oxford: Oxford University Press, ISBN 978-0-19-853171-5
Nicholas J. Higham, Handbook of writing for the mathematical sciences, SIAM. ISBN 0-89871-420-6, 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, ISBN 3-540-66957-4
Ribenboim, Paulo (1996), The New Book of Prime Number Records, New York: Springer, ISBN 0-387-94457-5
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., ISBN 0-19-853369-1
External linksedit
Wikimedia Commons has media related to Floor and ceiling functions.
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