{"history":[],"article":{"id":"fa787ed6-6ade-4b2d-a464-074338686ef1","title":"Floor and ceiling functions","slug":"floor-and-ceiling-functions","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **floor [function](https://bliptext.com/articles/function-mathematics)** is the [function](https://bliptext.com/articles/function-mathematics) that takes as input a [real number](https://bliptext.com/articles/real-number) x, and gives as output the greatest [integer](https://bliptext.com/articles/integer) less than or equal to x, denoted\n⌊x⌋\nor\nfloor(x)\n. Similarly, the **ceiling function** maps x to the least integer greater than or equal to\nx\n, denoted\n⌈x⌉\nor\nceil(x)\n.\nFor example, for floor:\n⌊2.4⌋ = 2\n,\n⌊−2.4⌋ = −3\n, and for ceiling:\n⌈2.4⌉ = 3\n, and\n⌈−2.4⌉ = −2\n.\nThe floor of x is also called the **integral part**, **integer part**, **greatest integer**, or **entier** of x, and was historically denoted\n[x]\n(among other notations).\nHowever, the same term, integer part, is also used for [truncation](https://bliptext.com/articles/truncation) towards zero, which differs from the floor function for negative numbers.\nFor n an integer,\n⌊n⌋ = ⌈n⌉ = n\n.\nAlthough\nfloor(x+1)\nand\nceil(x)\nproduce graphs that appear exactly alike, they are not the same when the value of x is an exact integer.\nFor example, when x=2.0001;\n⌊2.0001+1⌋ = ⌈2.0001⌉ = 3\n. However, if x=2, then\n⌊2+1⌋ = 3\n, while\n⌈2⌉ = 2\n.\n\n# Notation\nThe integral part or integer part of a number (partie entière in the original) was first defined in 1798 by [Adrien-Marie Legendre](https://bliptext.com/articles/adrien-marie-legendre) in his proof of the [Legendre's formula](https://bliptext.com/articles/legendre-s-formula).\n[Carl Friedrich Gauss](https://bliptext.com/articles/carl-friedrich-gauss) introduced the square bracket notation\n[x]\nin his third proof of [quadratic reciprocity](https://bliptext.com/articles/quadratic-reciprocity) (1808).\nThis remained the standard in mathematics until [Kenneth E. Iverson](https://bliptext.com/articles/kenneth-e-iverson) introduced, in his 1962 book A Programming Language, the names \"floor\" and \"ceiling\" and the corresponding notations\n⌊x⌋\nand\n⌈x⌉\n. (Iverson used square brackets for a different purpose, the [Iverson bracket](https://bliptext.com/articles/iverson-bracket) notation.) Both notations are now used in mathematics, although Iverson's notation will be followed in this article.\nIn some sources, boldface or double brackets\n⟦x⟧\nare used for floor, and reversed brackets\n⟧x⟦\nor\n]x[\nfor ceiling.\nThe [fractional part](https://bliptext.com/articles/fractional-part) is the [sawtooth function](https://bliptext.com/articles/sawtooth-function), denoted by\n{x}\nfor real x and defined by the formula\n{x} = x − ⌊x⌋\nFor all x,\n0 ≤ {x} < 1\n.\nThese characters are provided in Unicode:\nIn the [LaTeX](https://bliptext.com/articles/latex) typesetting system, these symbols can be specified with the and commands in math mode.\nLaTeX has supported UTF-8 since 2018, so the Unicode characters can now be used directly.\nLarger versions are and.\n\n# Definition and properties\nGiven real numbers x and y, [integers](https://bliptext.com/articles/integer) m and n and the set of [integers](https://bliptext.com/articles/integer) \\mathbb{Z}, floor and ceiling may be defined by the equations\nSince there is exactly one integer in a [half-open interval](https://bliptext.com/articles/half-open-interval) of length one, for any real number x, there are unique integers m and n satisfying the equation\nwhere and may also be taken as the definition of floor and ceiling.\n\n## Equivalences\nThese formulas can be used to simplify expressions involving floors and ceilings.\nIn the language of [order theory](https://bliptext.com/articles/order-theory), the floor function is a [residuated mapping](https://bliptext.com/articles/residuated-mapping), that is, part of a [Galois connection](https://bliptext.com/articles/galois-connection): it is the upper adjoint of the function that embeds the integers into the reals.\nThese formulas show how adding an integer n to the arguments affects the functions:\nThe above are never true if n is not an integer; however, for every x and y, the following inequalities hold:\n\n## Monotonicity\nBoth floor and ceiling functions are [monotonically non-decreasing functions](https://bliptext.com/articles/monotonic-function):\n\n## Relations among the functions\nIt is clear from the definitions that\nIn fact, for integers n, both floor and ceiling functions are the [identity](https://bliptext.com/articles/identity-function):\nNegating the argument switches floor and ceiling and changes the sign:\nand:\nNegating the argument complements the fractional part:\nThe floor, ceiling, and fractional part functions are [idempotent](https://bliptext.com/articles/idempotence):\nThe result of nested floor or ceiling functions is the innermost function:\ndue to the identity property for integers.\n\n## Quotients\nIf m and n are integers and n ≠ 0,\nIf n is a positive integer\nIf m is positive\nFor m = 2 these imply\nMore generally, for positive m (See [Hermite's identity](https://bliptext.com/articles/hermite-s-identity))\nThe following can be used to convert floors to ceilings and vice versa (m positive)\nFor all m and n strictly positive integers:\nwhich, for positive and [coprime](https://bliptext.com/articles/coprime) m and n, reduces to\nand similarly for the ceiling and fractional part functions (still for positive and [coprime](https://bliptext.com/articles/coprime) m and n),\nSince the right-hand side of the general case is symmetrical in m and n, this implies that\nMore generally, if m and n are positive,\nThis is sometimes called a [reciprocity law](https://bliptext.com/articles/).\nDivision by positive integers gives rise to an interesting and sometimes useful property.\nAssuming m,n >0,\nSimilarly,\nIndeed,\nkeeping in mind that\nThe second equivalence involving the ceiling function can be proved similarly.\n\n## Nested divisions\nFor positive integer n, and arbitrary real numbers m,x:\n\n## Continuity and series expansions\nNone of the functions discussed in this article are [continuous](https://bliptext.com/articles/continuous-function), but all are [piecewise linear](https://bliptext.com/articles/piecewise-linear-function): the functions, , and \\{ x\\} have discontinuities at the integers.\nis [upper semi-continuous](https://bliptext.com/articles/semi-continuity) and and \\{ x\\} are lower semi-continuous.\nSince none of the functions discussed in this article are continuous, none of them have a [power series](https://bliptext.com/articles/power-series) expansion.\nSince floor and ceiling are not periodic, they do not have uniformly convergent [Fourier series](https://bliptext.com/articles/fourier-series) expansions.\nThe fractional part function has Fourier series expansion\nfor x not an integer.\nAt 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.\nAt points of continuity the series converges to the true value.\nUsing the formula gives\nfor x not an integer.\n\n# Applications\n\n## Mod operator\nFor an integer x and a positive integer y, the [modulo operation](https://bliptext.com/articles/modulo-operation), denoted by x mod y, gives the value of the remainder when x is divided by y.\nThis definition can be extended to real x and y, y ≠ 0, by the formula\nThen it follows from the definition of floor function that this extended operation satisfies many natural properties.\nNotably, x mod y is always between 0 and y, i.e.,\nif y is positive,\nand if y is negative,\n\n## Quadratic reciprocity\nGauss's third proof of [quadratic reciprocity](https://bliptext.com/articles/quadratic-reciprocity), as modified by Eisenstein, has two basic steps.\nLet p and q be distinct positive odd prime numbers, and let\nFirst, [Gauss's lemma](https://bliptext.com/articles/gauss-s-lemma-number-theory) is used to show that the [Legendre symbols](https://bliptext.com/articles/legendre-symbol) are given by\nThe second step is to use a [geometric](https://bliptext.com/articles/geometric-series) argument to show that\nCombining these formulas gives quadratic reciprocity in the form\nThere are formulas that use floor to express the quadratic character of small numbers mod odd primes p:\n\n## Rounding\nFor an arbitrary real number x, [rounding](https://bliptext.com/articles/rounding) x to the nearest integer with [tie breaking](https://bliptext.com/articles/rounding) towards positive infinity is given by ; [rounding](https://bliptext.com/articles/rounding) towards negative infinity is given as.\nIf tie-breaking is away from 0, then the rounding function is (see [sign function](https://bliptext.com/articles/sign-function)), and [rounding towards even](https://bliptext.com/articles/rounding) can be expressed with the more cumbersome, which is the above expression for rounding towards positive infinity minus an [integrality](https://bliptext.com/articles/integer) [indicator](https://bliptext.com/articles/indicator-function) for.\nRounding a [real number](https://bliptext.com/articles/real-number) x to the nearest integer value forms a very basic type of [quantizer](https://bliptext.com/articles/quantization-signal-processing) – a uniform one.\nA typical (mid-tread) uniform quantizer with a quantization step size equal to some value \\Delta can be expressed as\n\n## Number of digits\nThe number of digits in [base](https://bliptext.com/articles/base-exponentiation) b of a positive integer k is\n\n## Number of strings without repeated characters\nThe number of possible [strings](https://bliptext.com/articles/string-computer-science) of arbitrary length that doesn't use any character twice is given by\nwhere:\nn\n> 0 is the number of letters in the alphabet (e.g., 26 in [English](https://bliptext.com/articles/english-language))\nk\nthat don't use any character twice.\nn\n! denotes the [factorial](https://bliptext.com/articles/factorial) of\nn\ne\n= 2.718... is [Euler's number](https://bliptext.com/articles/euler-s-number)\nFor\nn\n= 26, this comes out to 1096259850353149530222034277.\n\n## Factors of factorials\nLet n be a positive integer and p a positive prime number.\nThe exponent of the highest power of p that divides n!\nis given by a version of [Legendre's formula](https://bliptext.com/articles/legendre-s-formula)\nwhere is the way of writing n in base p.\nThis is a finite sum, since the floors are zero when pk > n.\n\n## Beatty sequence\nThe [Beatty sequence](https://bliptext.com/articles/beatty-sequence) shows how every positive [irrational number](https://bliptext.com/articles/irrational-number) gives rise to a partition of the [natural numbers](https://bliptext.com/articles/natural-number) into two sequences via the floor function.\n\n## Euler's constant (γ)\nThere are formulas for [Euler's constant](https://bliptext.com/articles/euler-mascheroni-constant) γ = 0.57721 56649 ... that involve the floor and ceiling, e.g.\nand\n\n## Riemann zeta function (ζ)\nThe fractional part function also shows up in integral representations of the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function).\nIt is straightforward to prove (using integration by parts) that if \\varphi(x) is any function with a continuous derivative in the closed interval [a, b],\nLetting for [real part](https://bliptext.com/articles/real-part) of s greater than 1 and letting a and b be integers, and letting b approach infinity gives\nThis 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.\nFor s = σ + it in the critical strip 0 < σ < 1,\nIn 1947 [van der Pol](https://bliptext.com/articles/balthasar-van-der-pol) used this representation to construct an analogue computer for finding roots of the zeta function.\n\n## Formulas for prime numbers\nThe floor function appears in several formulas characterizing prime numbers.\nFor 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](https://bliptext.com/articles/if-and-only-if)\nOne may also give formulas for producing the prime numbers.\nFor example, let pn be the n-th prime, and for any integer r > 1, define the real number α by the sum\nThen\nA similar result is that there is a number θ = 1.3064...\n([Mills' constant](https://bliptext.com/articles/mills-constant)) with the property that\nare all prime.\nThere is also a number ω = 1.9287800...\nwith the property that\nare all prime.\nLet π(x) be the number of primes less than or equal to x.\nIt is a straightforward deduction from [Wilson's theorem](https://bliptext.com/articles/wilson-s-theorem) that\nAlso, if n ≥ 2,\nNone of the formulas in this section are of any practical use.\n\n## Solved problems\n[Ramanujan](https://bliptext.com/articles/ramanujan) submitted these problems to the Journal of the Indian Mathematical Society.\nIf n is a positive integer, prove that\n

\nSome generalizations to the above floor function identities have been proven.\n\n## Unsolved problem\nThe study of [Waring's problem](https://bliptext.com/articles/waring-s-problem) has led to an unsolved problem:\nAre there any positive integers k ≥ 6 such that\n[Mahler](https://bliptext.com/articles/kurt-mahler) has proved there can only be a finite number of such k; none are known.\n\n# Computer implementations\nIn most programming languages, the simplest method to convert a floating point number to an integer does not do floor or ceiling, but truncation.\nThe reason for this is historical, as the first machines used [ones' complement](https://bliptext.com/articles/ones-complement) and truncation was simpler to implement (floor is simpler in [two's complement](https://bliptext.com/articles/two-s-complement)).\n[FORTRAN](https://bliptext.com/articles/fortran) was defined to require this behavior and thus almost all processors implement conversion this way.\nSome 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.\nAn [arithmetic right-shift](https://bliptext.com/articles/arithmetic-shift) of a signed integer x by n is the same as.\nDivision 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.\nAssuming such shifts are \"premature optimization\" and replacing them with division can break software.\nMany programming languages (including [C](https://bliptext.com/articles/c-programming-language), [C](https://bliptext.com/articles/c-programming-language)++, [C](https://bliptext.com/articles/c-programming-language)#, [Java](https://bliptext.com/articles/java-programming-language), [Julia](https://bliptext.com/articles/julia-programming-language), [PHP](https://bliptext.com/articles/php), [R](https://bliptext.com/articles/r-programming-language), and [Python](https://bliptext.com/articles/python-programming-language) ) provide standard functions for floor and ceiling, usually called and , or less commonly.\nThe language [APL](https://bliptext.com/articles/apl-programming-language) uses for floor.\nThe [J Programming Language](https://bliptext.com/articles/j-programming-language), a follow-on to APL that is designed to use standard keyboard symbols, uses for floor and for ceiling.\n[ALGOL](https://bliptext.com/articles/algol) uses for floor.\nIn [Microsoft Excel](https://bliptext.com/articles/microsoft-excel) the function rounds down rather than toward zero, while rounds toward zero, the opposite of what \"int\" and \"floor\" do in other languages.\nSince 2010 has been changed to error if the number is negative.\nThe [OpenDocument](https://bliptext.com/articles/opendocument) file format, as used by [OpenOffice.org](https://bliptext.com/articles/openoffice-org), [Libreoffice](https://bliptext.com/articles/libreoffice) and others, and both do floor, and has a third argument to reproduce Excel's earlier behavior.\n# [Int function from floating-point conversion in C | upload.wikimedia.org/wikipedia/commons/a/a9/Int///function.svg]\n\n# Citations","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:38:05.717Z","updated_at":"2025-01-25T17:38:05.717Z","search_vector":"'ceil':3 'floor':1 'function':4","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **floor [function](https://bliptext.com/articles/function-mathematics)** is the [function](https://bliptext.com/articles/function-mathematics) that takes as input a [real number](https://bliptext.com/articles/real-number) x, and gives as output the greatest [integer](https://bliptext.com/articles/integer) less than or equal to x, denoted\n⌊x⌋\nor\nfloor(x)\n. Similarly, the **ceiling function** maps x to the least integer greater than or equal to\nx\n, denoted\n⌈x⌉\nor\nceil(x)\n.\nFor example, for floor:\n⌊2.4⌋ = 2\n,\n⌊−2.4⌋ = −3\n, and for ceiling:\n⌈2.4⌉ = 3\n, and\n⌈−2.4⌉ = −2\n.\nThe floor of x is also called the **integral part**, **integer part**, **greatest integer**, or **entier** of x, and was historically denoted\n[x]\n(among other notations).\nHowever, the same term, integer part, is also used for [truncation](https://bliptext.com/articles/truncation) towards zero, which differs from the floor function for negative numbers.\nFor n an integer,\n⌊n⌋ = ⌈n⌉ = n\n.\nAlthough\nfloor(x+1)\nand\nceil(x)\nproduce graphs that appear exactly alike, they are not the same when the value of x is an exact integer.\nFor example, when x=2.0001;\n⌊2.0001+1⌋ = ⌈2.0001⌉ = 3\n. However, if x=2, then\n⌊2+1⌋ = 3\n, while\n⌈2⌉ = 2\n.\n\n# Notation\nThe integral part or integer part of a number (partie entière in the original) was first defined in 1798 by [Adrien-Marie Legendre](https://bliptext.com/articles/adrien-marie-legendre) in his proof of the [Legendre's formula](https://bliptext.com/articles/legendre-s-formula).\n[Carl Friedrich Gauss](https://bliptext.com/articles/carl-friedrich-gauss) introduced the square bracket notation\n[x]\nin his third proof of [quadratic reciprocity](https://bliptext.com/articles/quadratic-reciprocity) (1808).\nThis remained the standard in mathematics until [Kenneth E. Iverson](https://bliptext.com/articles/kenneth-e-iverson) introduced, in his 1962 book A Programming Language, the names \"floor\" and \"ceiling\" and the corresponding notations\n⌊x⌋\nand\n⌈x⌉\n. (Iverson used square brackets for a different purpose, the [Iverson bracket](https://bliptext.com/articles/iverson-bracket) notation.) Both notations are now used in mathematics, although Iverson's notation will be followed in this article.\nIn some sources, boldface or double brackets\n⟦x⟧\nare used for floor, and reversed brackets\n⟧x⟦\nor\n]x[\nfor ceiling.\nThe [fractional part](https://bliptext.com/articles/fractional-part) is the [sawtooth function](https://bliptext.com/articles/sawtooth-function), denoted by\n{x}\nfor real x and defined by the formula\n{x} = x − ⌊x⌋\nFor all x,\n0 ≤ {x} 0,\nSimilarly,\nIndeed,\nkeeping in mind that\nThe second equivalence involving the ceiling function can be proved similarly.\n\n## Nested divisions\nFor positive integer n, and arbitrary real numbers m,x:\n\n## Continuity and series expansions\nNone of the functions discussed in this article are [continuous](https://bliptext.com/articles/continuous-function), but all are [piecewise linear](https://bliptext.com/articles/piecewise-linear-function): the functions, , and \\{ x\\} have discontinuities at the integers.\nis [upper semi-continuous](https://bliptext.com/articles/semi-continuity) and and \\{ x\\} are lower semi-continuous.\nSince none of the functions discussed in this article are continuous, none of them have a [power series](https://bliptext.com/articles/power-series) expansion.\nSince floor and ceiling are not periodic, they do not have uniformly convergent [Fourier series](https://bliptext.com/articles/fourier-series) expansions.\nThe fractional part function has Fourier series expansion\nfor x not an integer.\nAt 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.\nAt points of continuity the series converges to the true value.\nUsing the formula gives\nfor x not an integer.\n\n# Applications\n\n## Mod operator\nFor an integer x and a positive integer y, the [modulo operation](https://bliptext.com/articles/modulo-operation), denoted by x mod y, gives the value of the remainder when x is divided by y.\nThis definition can be extended to real x and y, y ≠ 0, by the formula\nThen it follows from the definition of floor function that this extended operation satisfies many natural properties.\nNotably, x mod y is always between 0 and y, i.e.,\nif y is positive,\nand if y is negative,\n\n## Quadratic reciprocity\nGauss's third proof of [quadratic reciprocity](https://bliptext.com/articles/quadratic-reciprocity), as modified by Eisenstein, has two basic steps.\nLet p and q be distinct positive odd prime numbers, and let\nFirst, [Gauss's lemma](https://bliptext.com/articles/gauss-s-lemma-number-theory) is used to show that the [Legendre symbols](https://bliptext.com/articles/legendre-symbol) are given by\nThe second step is to use a [geometric](https://bliptext.com/articles/geometric-series) argument to show that\nCombining these formulas gives quadratic reciprocity in the form\nThere are formulas that use floor to express the quadratic character of small numbers mod odd primes p:\n\n## Rounding\nFor an arbitrary real number x, [rounding](https://bliptext.com/articles/rounding) x to the nearest integer with [tie breaking](https://bliptext.com/articles/rounding) towards positive infinity is given by ; [rounding](https://bliptext.com/articles/rounding) towards negative infinity is given as.\nIf tie-breaking is away from 0, then the rounding function is (see [sign function](https://bliptext.com/articles/sign-function)), and [rounding towards even](https://bliptext.com/articles/rounding) can be expressed with the more cumbersome, which is the above expression for rounding towards positive infinity minus an [integrality](https://bliptext.com/articles/integer) [indicator](https://bliptext.com/articles/indicator-function) for.\nRounding a [real number](https://bliptext.com/articles/real-number) x to the nearest integer value forms a very basic type of [quantizer](https://bliptext.com/articles/quantization-signal-processing) – a uniform one.\nA typical (mid-tread) uniform quantizer with a quantization step size equal to some value \\Delta can be expressed as\n\n## Number of digits\nThe number of digits in [base](https://bliptext.com/articles/base-exponentiation) b of a positive integer k is\n\n## Number of strings without repeated characters\nThe number of possible [strings](https://bliptext.com/articles/string-computer-science) of arbitrary length that doesn't use any character twice is given by\nwhere:\nn\n> 0 is the number of letters in the alphabet (e.g., 26 in [English](https://bliptext.com/articles/english-language))\nk\nthat don't use any character twice.\nn\n! denotes the [factorial](https://bliptext.com/articles/factorial) of\nn\ne\n= 2.718... is [Euler's number](https://bliptext.com/articles/euler-s-number)\nFor\nn\n= 26, this comes out to 1096259850353149530222034277.\n\n## Factors of factorials\nLet n be a positive integer and p a positive prime number.\nThe exponent of the highest power of p that divides n!\nis given by a version of [Legendre's formula](https://bliptext.com/articles/legendre-s-formula)\nwhere is the way of writing n in base p.\nThis is a finite sum, since the floors are zero when pk > n.\n\n## Beatty sequence\nThe [Beatty sequence](https://bliptext.com/articles/beatty-sequence) shows how every positive [irrational number](https://bliptext.com/articles/irrational-number) gives rise to a partition of the [natural numbers](https://bliptext.com/articles/natural-number) into two sequences via the floor function.\n\n## Euler's constant (γ)\nThere are formulas for [Euler's constant](https://bliptext.com/articles/euler-mascheroni-constant) γ = 0.57721 56649 ... that involve the floor and ceiling, e.g.\nand\n\n## Riemann zeta function (ζ)\nThe fractional part function also shows up in integral representations of the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function).\nIt is straightforward to prove (using integration by parts) that if \\varphi(x) is any function with a continuous derivative in the closed interval [a, b],\nLetting for [real part](https://bliptext.com/articles/real-part) of s greater than 1 and letting a and b be integers, and letting b approach infinity gives\nThis 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.\nFor s = σ + it in the critical strip 0 1, define the real number α by the sum\nThen\nA similar result is that there is a number θ = 1.3064...\n([Mills' constant](https://bliptext.com/articles/mills-constant)) with the property that\nare all prime.\nThere is also a number ω = 1.9287800...\nwith the property that\nare all prime.\nLet π(x) be the number of primes less than or equal to x.\nIt is a straightforward deduction from [Wilson's theorem](https://bliptext.com/articles/wilson-s-theorem) that\nAlso, if n ≥ 2,\nNone of the formulas in this section are of any practical use.\n\n## Solved problems\n[Ramanujan](https://bliptext.com/articles/ramanujan) submitted these problems to the Journal of the Indian Mathematical Society.\nIf n is a positive integer, prove that\n \n \n \n Some generalizations to the above floor function identities have been proven.\n\n## Unsolved problem\nThe study of [Waring's problem](https://bliptext.com/articles/waring-s-problem) has led to an unsolved problem:\nAre there any positive integers k ≥ 6 such that\n[Mahler](https://bliptext.com/articles/kurt-mahler) has proved there can only be a finite number of such k; none are known.\n\n# Computer implementations\nIn most programming languages, the simplest method to convert a floating point number to an integer does not do floor or ceiling, but truncation.\nThe reason for this is historical, as the first machines used [ones' complement](https://bliptext.com/articles/ones-complement) and truncation was simpler to implement (floor is simpler in [two's complement](https://bliptext.com/articles/two-s-complement)).\n[FORTRAN](https://bliptext.com/articles/fortran) was defined to require this behavior and thus almost all processors implement conversion this way.\nSome 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.\nAn [arithmetic right-shift](https://bliptext.com/articles/arithmetic-shift) of a signed integer x by n is the same as.\nDivision 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.\nAssuming such shifts are \"premature optimization\" and replacing them with division can break software.\nMany programming languages (including [C](https://bliptext.com/articles/c-programming-language), [C](https://bliptext.com/articles/c-programming-language)++, [C](https://bliptext.com/articles/c-programming-language)#, [Java](https://bliptext.com/articles/java-programming-language), [Julia](https://bliptext.com/articles/julia-programming-language), [PHP](https://bliptext.com/articles/php), [R](https://bliptext.com/articles/r-programming-language), and [Python](https://bliptext.com/articles/python-programming-language) ) provide standard functions for floor and ceiling, usually called and , or less commonly.\nThe language [APL](https://bliptext.com/articles/apl-programming-language) uses for floor.\nThe [J Programming Language](https://bliptext.com/articles/j-programming-language), a follow-on to APL that is designed to use standard keyboard symbols, uses for floor and for ceiling.\n[ALGOL](https://bliptext.com/articles/algol) uses for floor.\nIn [Microsoft Excel](https://bliptext.com/articles/microsoft-excel) the function rounds down rather than toward zero, while rounds toward zero, the opposite of what \"int\" and \"floor\" do in other languages.\nSince 2010 has been changed to error if the number is negative.\nThe [OpenDocument](https://bliptext.com/articles/opendocument) file format, as used by [OpenOffice.org](https://bliptext.com/articles/openoffice-org), [Libreoffice](https://bliptext.com/articles/libreoffice) and others, and both do floor, and has a third argument to reproduce Excel's earlier behavior.\n# [Int function from floating-point conversion in C | upload.wikimedia.org/wikipedia/commons/a/a9/Int///function.svg]\n\n# Citations","revisionCount":0},"hasMore":false,"nextCursor":null}

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