Floor and ceiling functions

Last updated

Floor and ceiling functions
Floor function.svg
Floor function
Ceiling function.svg
Ceiling function

In mathematics and computer science, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted floor(x) or x. Similarly, the ceiling function maps x to the least integer greater than or equal to x, denoted ceil(x) or x. [1]


For example, ⌊2.4⌋ = 2, 2.4⌋ = 3, ⌈2.4⌉ = 3, and 2.4⌉ = 2.

The integral part or integer part of x, often denoted [x] is usually defined as the x if x is nonnegative, and x otherwise.[ citation needed ] For example, [2.4] = 2 and [2.4] = 2. The operation of truncation generalizes this to a specified number of digits: truncation to zero significant digits is the same as the integer part.

Some authors define the integer part as the floor regardless of the sign of x, using a variety of notations for this. [2]

For n an integer, n⌋ = ⌈n⌉ = [n] = n.

xFloor xCeiling x Fractional part {x}


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] Both notations are now used in mathematics, [7] 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. [8] [9] Sometimes [x] is taken to mean the round-toward-zero function.[ citation needed ]

The fractional part is the sawtooth function, denoted by {x} for real x and defined by the formula

{x} = x - ⌊x [10]

For all x,

0 ≤ {x} < 1

These characters are provided in Unicode:

In the LaTeX typesetting system, these symbols can be specified with the \lfloor, \rfloor, \lceil and \rceil commands in math mode, and extended in size using \left\lfloor, \right\rfloor, \left\lceil and \right\rceil as needed.

Definition and properties

Given real numbers x and y, integers k, m, n and the set of integers , 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

where  and  may also be taken as the definition of floor and ceiling.


These 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 is the upper adjoint of the function that embeds the integers into the reals.

These formulas show how adding integers to the arguments affects the functions:

The above are never true if n is not an integer; however, for every x and y, the following inequalities hold:


Both floor and ceiling functions are the monotonically non-decreasing function:

Relations among the functions

It is clear from the definitions that

  with equality if and only if x is an integer, i.e.

In fact, for integers n, both floor and ceiling functions are the identity:

Negating the argument switches floor and ceiling and changes the sign:


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:

due to the identity property for integers.


If m and n are integers and n ≠ 0,

If n is a positive integer [12]

If m is positive [13]

For m = 2 these imply

More generally, [14] for positive m (See Hermite's identity)

The following can be used to convert floors to ceilings and vice versa (m positive) [15]

For all m and n strictly positive integers: [16]

which, for positive and coprime m and n, reduces to

and similarly for the ceiling function (still for positive and coprime m and n),

Since the right-hand side of the general case is symmetrical in m and n, this implies that

More generally, if m and n are positive,

This is sometimes called a reciprocity law. [17]

Nested divisions

For positive integer n, and arbitrary real numbers m,x: [18]

Continuity and series expansions

None of the functions discussed in this article are continuous, but all are piecewise linear: the functions , , and have discontinuities at the integers.

  is upper semi-continuous and    and   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 [19]

for x not 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: 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.

Using the formula floor(x) = x {x} gives

for x not an integer.


Mod operator

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,

and if y is negative,

Quadratic reciprocity

Gauss's third proof of quadratic reciprocity, as modified by Eisenstein, has two basic steps. [20] [21]

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


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: [22]


For an arbitrary real number , rounding to the nearest integer with tie breaking towards positive infinity is given by ; rounding towards negative infinity is given as .

If tie-breaking is away from 0, then the rounding function is (see sign function), and rounding towards even can be expressed with the more cumbersome , which is the above expression for rounding towards positive infinity minus an integrality indicator for .

Number of digits

The number of digits in base b of a positive integer k is

Number of strings without repeated characters

The number of possible strings of arbitrary length that don't use any character twice is given by [23] [ better source needed ]


For n = 26, this comes out to 1096259850353149530222034277.

Factors of factorials

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 [24]

where is the way of writing n in base p. This is a finite sum, since the floors are zero when pk > n.

Beatty sequence

The Beatty sequence shows how every positive irrational number gives rise to a partition of the natural numbers into two sequences via the floor function. [25]

Euler's constant (γ)

There are formulas for Euler's constant γ = 0.57721 56649 ... that involve the floor and ceiling, e.g. [26]


Riemann zeta function (ζ)

The fractional part function also shows up in integral representations of the Riemann zeta function. It is straightforward to prove (using integration by parts) [27] that if is any function with a continuous derivative in the closed interval [a, b],

Letting 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 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]

For s = σ + it in the critical strip 0 < σ < 1,

In 1947 van der Pol used this representation to construct an analogue computer for finding roots of the zeta function. [29]

Formulas for prime numbers

The floor function appears in several formulas characterizing prime numbers. For example, since 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 [30]

One may also give formulas for producing the prime numbers. For example, let pn be the n-th prime, and for any integer r > 1, define the real number α by the sum

Then [31]

A similar result is that there is a number θ = 1.3064... (Mills' constant) with the property that

are all prime. [32]

There is also a number ω = 1.9287800... with the property that

are all prime. [32]

Let π(x) be the number of primes less than or equal to x. It is a straightforward deduction from Wilson's theorem that [33]

Also, if n ≥ 2, [34]

None of the formulas in this section are of any practical use. [35] [36]

Solved problems

Ramanujan submitted these problems to the Journal of the Indian Mathematical Society. [37]

If n is a positive integer, prove that

Unsolved problem

The study of Waring's problem has led to an unsolved problem:

Are there any positive integers k ≥ 6 such that [38]


Mahler [39] has proved there can only be a finite number of such k; none are known.

Computer implementations

Int function from floating-point conversion in C Int function.svg
Int function from floating-point conversion in C

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 ]

A bit-wise right-shift of a signed integer by is 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 (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 floor function is implemented as INT (which rounds down rather than toward zero). [52] The command FLOOR in earlier versions would round toward zero, effectively the opposite of what "int" and "floor" do in other languages. Since 2010 FLOOR has been fixed to round down, with extra arguments that can reproduce previous behavior. [53] The OpenDocument file format, as used by OpenOffice.org, Libreoffice and others, uses the same function names; INT does floor [54] and FLOOR has a third argument that can make it round toward zero. [55]

See also


  1. Graham, Knuth, & Patashnik, Ch. 3.1
  2. 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
  3. Lemmermeyer, pp. 10, 23.
  4. e.g. Cassels, Hardy & Wright, and Ribenboim use Gauss's notation, Graham, Knuth & Patashnik, and Crandall & Pomerance use Iverson's.
  5. Iverson, p. 12.
  6. Higham, p. 25.
  7. See the Wolfram MathWorld article.
  8. Mathwords: Floor Function.
  9. Mathwords: Ceiling Function
  10. Graham, Knuth, & Patashnik, p. 70.
  11. Graham, Knuth, & Patashink, Ch. 3
  12. Graham, Knuth, & Patashnik, p. 73
  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. Graham, Knuth, & Patashnik, p. 94
  18. Graham, Knuth, & Patashnik, p. 71, apply theorem 3.10 with x/m as input and the division by n as function
  19. Titchmarsh, p. 15, Eq. 2.1.7
  20. Lemmermeyer, § 1.4, Ex. 1.32–1.33
  21. Hardy & Wright, §§ 6.11–6.13
  22. Lemmermeyer, p. 25
  23. OEIS sequenceA000522(Total number of arrangements of a set with n elements: a(n) = Sum_{k=0..n} n!/k!.) (See Formulas.)
  24. Hardy & Wright, Th. 416
  25. Graham, Knuth, & Patashnik, pp. 77–78
  26. These formulas are from the Wikipedia article Euler's constant, which has many more.
  27. Titchmarsh, p. 13
  28. Titchmarsh, pp.14–15
  29. Crandall & Pomerance, p. 391
  30. 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 .
  31. Hardy & Wright, § 22.3
  32. 1 2 Ribenboim, p. 186
  33. Ribenboim, p. 181
  34. Crandall & Pomerance, Ex. 1.4, p. 46
  35. 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 ... "
  36. 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."
  37. Ramanujan, Question 723, Papers p. 332
  38. Hardy & Wright, p. 337
  39. Mahler, Kurt (1957). "On the fractional parts of the powers of a rational number II". Mathematika . 4: 122–124. doi:10.1112/S0025579300001170.
  40. "C++ reference of floor function" . Retrieved 5 December 2010.
  41. "C++ reference of ceil function" . Retrieved 5 December 2010.
  42. dotnet-bot. "Math.Floor Method (System)". docs.microsoft.com. Retrieved 28 November 2019.
  43. dotnet-bot. "Math.Ceiling Method (System)". docs.microsoft.com. Retrieved 28 November 2019.
  44. "Math (Java SE 9 & JDK 9 )". docs.oracle.com. Retrieved 20 November 2018.
  45. "Math (Java SE 9 & JDK 9 )". docs.oracle.com. Retrieved 20 November 2018.
  46. "PHP manual for ceil function" . Retrieved 18 July 2013.
  47. "PHP manual for floor function" . Retrieved 18 July 2013.
  48. "R: Rounding of Numbers".
  49. "Python manual for math module" . Retrieved 18 July 2013.
  50. Sullivan, p. 86.
  51. "Vocabulary". J Language. Retrieved 6 September 2011.
  52. "INT function" . Retrieved 29 October 2021.
  53. "FLOOR function" . Retrieved 29 October 2021.
  54. "Documentation/How Tos/Calc: INT function" . Retrieved 29 October 2021.
  55. "Documentation/How Tos/Calc: FLOOR function" . Retrieved 29 October 2021.

Related Research Articles

Golomb coding is a lossless data compression method using a family of data compression codes invented by Solomon W. Golomb in the 1960s. Alphabets following a geometric distribution will have a Golomb code as an optimal prefix code, making Golomb coding highly suitable for situations in which the occurrence of small values in the input stream is significantly more likely than large values.

Eulers constant Relates logarithm and harmonic series

Euler's constant is a mathematical constant usually denoted by the lowercase Greek letter gamma.

In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.

In mathematics and computer science, truncation is limiting the number of digits right of the decimal point.

In computer science, the Akra–Bazzi method, or Akra–Bazzi theorem, is used to analyze the asymptotic behavior of the mathematical recurrences that appear in the analysis of divide and conquer algorithms where the sub-problems have substantially different sizes. It is a generalization of the master theorem for divide-and-conquer recurrences, which assumes that the sub-problems have equal size. It is named after mathematicians Mohamad Akra and Louay Bazzi.

Toom–Cook, sometimes known as Toom-3, named after Andrei Toom, who introduced the new algorithm with its low complexity, and Stephen Cook, who cleaned the description of it, is a multiplication algorithm for large integers.

Farey sequence

In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size.

The fractional part or decimal part of a non‐negative real number is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than x, called floor of x or , its fractional part can be written as:

In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. No such formula which is efficiently computable is known. A number of constraints are known, showing what such a "formula" can and cannot be.

In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another.

In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iverson bracket of the statement x = y. It maps any statement to a function of the free variables in that statement. This function is defined to take the value 1 for the values of the variables for which the statement is true, and takes the value 0 otherwise. It is generally denoted by putting the statement inside square brackets:

Bijective numeration is any numeral system in which every non-negative integer can be represented in exactly one way using a finite string of digits. The name refers to the bijection that exists in this case between the set of non-negative integers and the set of finite strings using a finite set of symbols.

Empirical distribution function Distribution function associated with the empirical measure of a sample

In statistics, an empirical distribution function is the distribution function associated with the empirical measure of a sample. This cumulative distribution function is a step function that jumps up by 1/n at each of the n data points. Its value at any specified value of the measured variable is the fraction of observations of the measured variable that are less than or equal to the specified value.

The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers such that

A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of Euclidean division. Some are applied by hand, while others are employed by digital circuit designs and software.

In mathematics, Hermite's identity, named after Charles Hermite, gives the value of a summation involving the floor function. It states that for every real number x and for every positive integer n the following identity holds:

In mathematics, a Beatty sequence is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about them in 1926.

In statistics, the generalized Marcum Q-function of order is defined as

In number theory, the prime omega functions and count the number of prime factors of a natural number Thereby counts each distinct prime factor, whereas the related function counts the total number of prime factors of honoring their multiplicity. That is, if we have a prime factorization of of the form for distinct primes , then the respective prime omega functions are given by and . These prime factor counting functions have many important number theoretic relations.

In computer science, merge-insertion sort or the Ford–Johnson algorithm is a comparison sorting algorithm published in 1959 by L. R. Ford Jr. and Selmer M. Johnson. It uses fewer comparisons in the worst case than the best previously known algorithms, binary insertion sort and merge sort, and for 20 years it was the sorting algorithm with the fewest known comparisons. Although not of practical significance, it remains of theoretical interest in connection with the problem of sorting with a minimum number of comparisons. The same algorithm may have also been independently discovered by Stanisław Trybuła and Czen Ping.