Meta tags:
Headings (most frequently used words):
modulo, contents, variants, of, the, definition, notation, common, pitfalls, performance, issues, properties, identities, in, programming, languages, generalizations, see, also, notes, references, external, links, with, offset, implementing, other, definitions, using, truncation,
Text of the page (most frequently used words):
yes (274), the (194), mod (142), truncated (121), retrieved (106), and (75), 2026 (67), floored (66), remainder (58), #modulo (46), java (43), org (40), rem (37), com (36), for (34), programming (31), division (31), docs (30), displaystyle (29), language (28), api (28), euclidean (28), frac (27), standard (25), left (25), right (25), reference (24), languages (22), math (22), platform (22), that (22), documentation (21), function (21), oracle (21), edition (21), are (19), 2021 (18), negative (18), same (18), edit (17), iso (17), operation (16), defined (16), quotient (16), this (15), integer (15), lang (15), fmod (15), which (15), 2022 (14), iec (14), lfloor (14), rfloor (14), tcl (13), positive (13), rounded (13), dividend (13), operators (12), www (12), denom (12), using (11), with (11), both (11), some (11), sign (11), definition (11), long (11), arithmetic (10), operator (10), sec (10), where (10), int (10), common (10), has (10), when (10), but (10), see (10), thus (10), divisor (10), page (9), operations (9), dev (9), then (9), not (9), other (9), sgn (9), non (8), from (8), core (8), mathematical (8), functions (8), developer (8), expressions (8), also (8), basic (8), two (8), div (8), definitions (8), can (8), wikipedia (7), numbers (7), specification (7), 2024 (7), microsoft (7), undefined (7), one (7), dart (7), clojure (7), such (7), number (7), round (7), operatorname (7), text (6), use (6), example (6), modular (6), php (6), manual (6), apple (6), perl (6), binary (6), mathematics (6), pkg (6), big (6), properties (6), values (6), result (6), expression (6), implementation (6), these (6), numer (6), offset (6), constant (6), equation (6), rounding (6), trunc (6), search (5), about (5), may (5), agree (5), wolfram (5), version (5), commands (5), expr (5), ansi (5), standards (5), racket (5), python (5), std (5), ocaml (5), library (5), help (5), kotlin (5), cases (5), integers (5), package (5), godot (5), fortran (5), class (5), 2012 (5), ieeeremainder (5), systems (5), uses (5), many (5), ldiv_t (5), return (5), textstyle (5), notation (5), zero (5), signed (5), unsigned (5), ceiling (5), because (5), according (5), mathbb (5), contents (4), code (4), links (4), 2020 (4), mathop (4), national (4), institute (4), information (4), technology (4), command (4), rust (4), posix (4), nim (4), archived (4), original (4), engine (4), pdf (4), operands (4), openeuphoria (4), returns (4), c99 (4), like (4), pascal (4), define (4), bmod (4), between (4), general (4), all (4), following (4), case (4), have (4), sql (4), lceil (4), rceil (4), aligned (4), computing (4), divided (4), hide (4), move (4), sidebar (4), toggle (3), table (3), view (3), under (3), terms (3), computer (3), multiplication (3), april (3), system (3), zig (3), mathfunc (3), american (3), pubs (3), opengroup (3), stdlib (3), nasm (3), support (3), after (3), matlab (3), julia (3), sides (3), data (3), types (3), globalscope (3), section (3), glsl (3), january (3), new (3), york (3), full (3), numeric (3), factor (3), github (3), euphoria (3), erlang (3), elm (3), elixir (3), method (3), css (3), style (3), coffeescript (3), learn (3), 1999 (3), 9899 (3), first (3), multiplicative (3), algol (3), ada (3), faster (3), boute (3), 127 (3), knuth (3), prime (3), examples (3), identity (3), leave (3), choices (3), here (3), quot (3), divmod (3), truncation (3), let (3), floor (3), generalizations (3), floormod (3), strictmath (3), lisp (3), inverse (3), side (3), performance (3), possible (3), always (3), used (3), expressed (3), bitwise (3), is_odd (3), bool (3), test (3), however (3), begin (3), end (3), down (3), cdot (3), article (3), tools (3), main (3), add (2), contact (2), privacy (2), policy (2), available (2), organization (2), was (2), categories (2), articles (2), short (2), description (2), wikidata (2), external (2), webassembly (2), solidity (2), 1998 (2), smalltalk (2), generic (2), numerics (2), 2018 (2), purebasic (2), net (2), perldoc (2), ieee (2), netwide (2), assembler (2), 118 (2), doi (2), mumps (2), mathworks (2), get (2), libreoffice (2), julialang (2), only (2), either (2), floating (2), point (2), haxe (2), haskell (2), 1987 (2), frink (2), manipulation (2), factorcode (2), fsharp (2), basics (2), hexdocs (2), 2025 (2), num (2), mozilla (2), cascading (2), sheets (2), mdn (2), jtc (2), 2023 (2), cobol (2), double (2), 1990 (2), magnitude (2), than (2), 2003 (2), nonnegative (2), international (2), august (2), awk (2), revised (2), report (2), adobe (2), actionscript (2), abap (2), july (2), power (2), 2001 (2), 2019 (2), raymond (2), acm (2), donald (2), residue (2), congruence (2), references (2), namespace (2), independently (2), rational (2), produces (2), includes (2), others (2), must (2), otherwise (2), behavior (2), including (2), notes (2), exponentiation (2), disambiguation (2), calculated (2), inline (2), truncating (2), part (2), generally (2), more (2), calculating (2), given (2), implementing (2), implemented (2), mathematica (2), true (2), what (2), now (2), take (2), related (2), follows (2), usual (2), useful (2), called (2), time (2), idiv (2), x86 (2), theorem (2), 125 (2), i64 (2), rem_s (2), i32 (2), rem_u (2), dividingby (2), 116 (2), 114 (2), scheme (2), lua (2), javascript (2), biginteger (2), ceilmod (2), remainderunsigned (2), various (2), hand (2), identities (2), optimizations (2), modulus (2), exist (2), optimization (2), implement (2), simple (2), will (2), equivalence (2), instead (2), alternative (2), might (2), special (2), hardware (2), assuming (2), issues (2), signs (2), odd (2), pitfalls (2), similar (2), any (2), calculators (2), falls (2), depends (2), even (2), promotes (2), satisfy (2), chosen (2), neq (2), representative (2), variants (2), being (2), appearance (2), upload (2), file (2), changes (2), history (2), read (2), english (2), log (2), create (2), account (2), donate (2), menu (2), topic, mobile, cookie, statement, statistics, developers, conduct, legal, safety, contacts, disclaimers, additional, apply, site, you, registered, trademark, profit, wikimedia, foundation, inc, creative, commons, attribution, sharealike, license, rendered, parsoid, last, edited, september, utc, hidden, matches, https, index, title, oldid, 1373722450, animation, cyclic, representation, tables, explanation, french, modulorama, research, center, rossberg, andreas, world, wide, web, consortium, built, soliditylang, incits, 319, r2002, shell, r6rs, f32, doc, project, quantumwriter, euclideanring, interface, 1003, browser, chapter, 1976, gaithersburg, dept, commerce, 6028, nbs, neill, joseph, autodesk, june, unlike, operates, well, cross, toolkit, predefined, classes, results, frinklang, dll, kernel, gnu, arbitrary, precision, calculator, dlang, lispworks, clhs, february, iii, 23271, infrastructure, cli, 1989, dotnet, bot, compute, required, 60559, value, nonzero, less, yields, second, electrotechnical, commission, standardization, 14882, 2005, tc2, 1973, ifip, wg2, algorithmic, multiplying, 8652, as3, sap, keyword, horvath, adam, archive, adaic, lrm, evaluation, peterson, doctor, october, forum, ask, 1992, press, usa, 144, 8321674, s2cid, 1854, 314490, hdl, 1145, 128861, 128862, transactions, 1972, addison, wesley, art, caldwell, chris, glossary, weisstein, eric, mathworld, 124, 121, defines, applied, inexact, excludes, 109, usually, machine, independent, exceptions, discussed, obey, fundamentally, broken, before, argument, order, reverses, computes, dividing, omega, alpha, boldface, distinguish, keywords, identifiers, permits, formats, capitalization, mathematically, infinite, inequality, satisfied, turn, angle, term, word, grew, out, approach, 1801, carl, gauss, vice, versa, combined, save, screen, space, logical, branches, ldivf, else, ldive, structure, reproduced, clarity, struct, typedef, ldiv, despite, elegance, much, find, based, leijen, provides, algorithms, divisions, show, particular, means, wanted, prove, remains, shown, obtaining, seen, done, there, does, seem, tentatively, just, clearly, corresponds, sometimes, lie, particularly, addition, provide, functionality, include, instructions, architecture, prover, 126, xbase, fprem1, fprem, assembly, visual, viml, vhdl, verilog, turing, torque, tcsh, 123, 122, 120, truncatingremainder, 119, swift, stata, real, 2011, 117, spin, snap, 115, mksh, bash, sensetalk, seed7, scratch, flmod0, flmod, mod0, scala, sas, 113, rem_euclid, 112, ruby, rpg, rexx, reason, realbasic, raku, 111, 110, 108, 107, 106, 105, 104, pure, 103, purescript, 102, prolog, progress, prc, powershell, pic, pro, 101, 100, 7185, 10206, occam, mod_float, delphi, object, objective, oberon, nasmx, modula, minitab, excel, maya, embedded, maxima, frem, mods, default, modp, maple, mathcad, liberty, logo, labview, ksh, typescript, bigdecimal, mathcontext, hlsl, ghc, fixed, groovy, fposmod, posmod, gdscript, gml, gamemaker, studio, forth, filemaker, remainderby, modby, eiffel, crystal, intermediate, coldfusion, clean, clarion, varies, autolisp, applescript, apl, ampl, coprime, distributive, denotes, relatively, due, fermat, little, factored, expanded, similarly, proofs, involving, require, diffie, hellman, key, exchange, cryptography, recognize, form, automatically, them, allowing, programmer, write, clearer, without, compromising, unless, type, whereas, compiler, devices, software, efficiently, forms, calculations, each, alternatives, alternatively, powers, correct, regardless, incorrect, false, inclined, equal, lead, surprising, mistakes, environments, lacking, three, above, button, relation, satisfies, opposite, its, within, boundaries, half, 754, equivalently, emphasis, added, say, mathrm, color, red, leq, implementations, significant, digits, toward, integral, still, leaves, ambiguity, occur, choice, determines, consecutive, quotients, choose, depending, give, divisors, c90, details, though, theory, quad, nearly, conditions, member, smallest, belongs, conventions, computers, ways, storing, representing, their, underlying, least, exactly, breaks, differ, how, although, typically, performed, allow, range, evaluates, while, would, evaluate, often, abbreviated, another, latter, computational, redirected, free, encyclopedia, item, projects, printable, download, print, export, switch, legacy, parser, shortened, url, cite, permanent, link, actions, talk, tiếng, việt, türkçe, svenska, српски, srpski, slovenščina, română, português, polski, norsk, bokmål, 한국어, қазақша, 日本語, italiano, íslenska, bahasa, indonesia, հայերեն, français, فارسی, español, deutsch, dansk, català, العربية, subsection, top, personal, pages, recent, community, portal, contribute, random, current, events, navigation, jump, content,
Text of the page (random words):
tion returns the remainder or signed remainder of a division after one number is divided by another the latter being called the modulus of the operation given two positive numbers a and n a modulo n often abbreviated as a mod n is the remainder of the euclidean division of a by n where a is the dividend and n is the divisor 1 for example the expression 5 mod 2 evaluates to 1 because 5 divided by 2 has a quotient of 2 and a remainder of 1 while 9 mod 3 would evaluate to 0 because 9 divided by 3 has a quotient of 3 and a remainder of 0 although typically performed with a and n both being integers many computing systems now allow other types of numeric operands the range of values for an integer modulo operation of n is 0 to n 1 a mod 1 is always 0 assuming a is an integer when exactly one of a or n is negative the basic definition breaks down and programming languages differ in how these values are defined variants of the definition edit in mathematics the result of the modulo operation is an equivalence class and any member of the class may be chosen as representative however the usual representative is the least positive residue the smallest non negative integer that belongs to that class i e the remainder of the euclidean division 2 however other conventions are possible computers and calculators have various ways of storing and representing numbers thus their definition of the modulo operation depends on the programming language or the underlying hardware in nearly all computing systems the quotient q and the remainder r of a divided by n 0 displaystyle n neq 0 satisfy the following conditions q z a n q r n 0 r n displaystyle begin aligned q in mathbb z a nq r quad n neq 0 r n end aligned 1 this still leaves a sign ambiguity if the remainder is non zero two possible choices for the remainder occur one negative and the other positive that choice determines which of the two consecutive quotients must be used to satisfy equation 1 in number theory the positive remainder is always chosen but in computing programming languages choose depending on the language and the signs of a or n a standard pascal and algol 68 for example give a positive remainder or 0 even for negative divisors and some programming languages such as c90 leave it to the implementation when either of n or a is negative see the table under in programming languages for details some systems leave a modulo 0 undefined though others define it as a quotient q and remainder r as functions of dividend a using truncated division many implementations use truncated division for which the quotient is defined by q trunc a n displaystyle q operatorname trunc left frac a n right where trunc displaystyle operatorname trunc is the integral part function rounding toward zero i e the truncation to zero significant digits thus according to equation 1 the remainder has the same sign as the dividend a so can take 2 n 1 values r a n trunc a n displaystyle r a n operatorname trunc left frac a n right quotient and remainder using floored division donald knuth 3 promotes floored division for which the quotient is defined by q a n displaystyle q left lfloor frac a n right rfloor where displaystyle lfloor rfloor is the floor function rounding down thus according to equation 1 the remainder has the same sign as the divisor n r a n a n displaystyle r a n left lfloor frac a n right rfloor quotient and remainder using euclidean division raymond t boute 4 promotes euclidean division for which the non negative remainder r 0 1 2 displaystyle r in 0 1 2 is defined by r a n q s u c h t h a t 0 r n displaystyle r a nq mathrm such that color red 0 leq r n emphasis added under this definition we can say the following about the quotient q displaystyle q q a r n z sgn n a r n sgn n a n r n sgn n a n displaystyle begin aligned q frac a r n in mathbb z text sgn n cdot frac a r n text sgn n cdot left frac a n frac r n right text sgn n cdot left lfloor frac a left n right right rfloor end aligned where sgn is the sign function displaystyle lfloor rfloor is the floor function rounding down and a n q displaystyle frac a n in mathbb q r n q displaystyle frac r n in mathbb q are rational numbers equivalently one may instead define the quotient q z displaystyle q in mathbb z as follows q sgn n a n a n if n 0 a n if n 0 displaystyle q operatorname sgn n left lfloor frac a left n right right rfloor begin cases left lfloor frac a n right rfloor text if n 0 left lceil frac a n right rceil text if n 0 end cases where displaystyle lceil rceil is the ceiling function rounding up thus according to equation 1 the remainder r displaystyle r is non negative r a n q a n a n displaystyle r a nq a n left lfloor frac a left n right right rfloor quotient and remainder using rounded division common lisp and ieee 754 use rounded division for which the quotient is defined by q round a n displaystyle q operatorname round left frac a n right where round is the round function rounding half to even thus according to equation 1 the remainder falls between n 2 displaystyle frac n 2 and n 2 displaystyle frac n 2 and its sign depends on which side of zero it falls to be within these boundaries r a n round a n displaystyle r a n operatorname round left frac a n right quotient and remainder using ceiling division common lisp also uses ceiling division for which the quotient is defined by q a n displaystyle q left lceil frac a n right rceil where is the ceiling function rounding up thus according to equation 1 the remainder has the opposite sign of that of the divisor r a n a n displaystyle r a n left lceil frac a n right rceil if both the dividend and divisor are positive then the truncated floored and euclidean definitions agree if the dividend is positive and the divisor is negative then the truncated and euclidean definitions agree if the dividend is negative and the divisor is positive then the floored and euclidean definitions agree if both the dividend and divisor are negative then the truncated and floored definitions agree however truncated division satisfies the identity a b a b a b displaystyle a b a b a b 5 6 notation edit this section is about the binary mod operation for the mod m notation see congruence relation some calculators have a mod function button and many programming languages have a similar function expressed as mod a n for example some also support expressions that use mod or mod as a modulo or remainder operator such as a n or a mod n for environments lacking a similar function any of the three definitions above can be used common pitfalls edit when the result of a modulo operation has the sign of the dividend truncated definition it can lead to surprising mistakes for example to test if an integer is odd one might be inclined to test if the remainder by 2 is equal to 1 bool is_odd int n return n 2 1 but in a language where modulo has the sign of the dividend that is incorrect because when n the dividend is negative and odd n mod 2 returns 1 and the function returns false one correct alternative is to test that the remainder is not 0 because remainder 0 is the same regardless of the signs bool is_odd int n return n 2 0 or with the binary arithmetic bool is_odd int n return n 1 performance issues edit modulo operations might be implemented such that a division with a remainder is calculated each time for special cases on some hardware faster alternatives exist for example the modulo of powers of 2 can alternatively be expressed as a bitwise and operation assuming x is a positive integer or using a non truncating definition x 2 n x 2 n 1 examples x 2 x 1 x 4 x 3 x 8 x 7 in devices and software that implement bitwise operations more efficiently than modulo these alternative forms can result in faster calculations 7 compiler optimizations may recognize expressions of the form expression constant where constant is a power of two and automatically implement them as expression constant 1 allowing the programmer to write clearer code without compromising performance this simple optimization is not possible for languages in which the result of the modulo operation has the sign of the dividend including c unless the dividend is of an unsigned integer type this is because if the dividend is negative the modulo will be negative whereas expression constant 1 will always be positive for these languages the equivalence x 2 n x 0 x 2 n 1 x 2 n 1 has to be used instead expressed using bitwise or not and and operations optimizations for general constant modulus operations also exist by calculating the division first using the constant divisor optimization properties identities edit see also modular arithmetic properties some modulo operations can be factored or expanded similarly to other mathematical operations this may be useful in cryptography proofs such as the diffie hellman key exchange the properties involving multiplication division and exponentiation generally require that a and n are integers identity a mod n mod n a mod n n x mod n 0 for all positive integer values of x if p is a prime number which is not a divisor of b then ab p 1 mod p a mod p due to fermat s little theorem inverse a mod n a mod n mod n 0 b 1 mod n denotes the modular multiplicative inverse which is defined if and only if b and n are relatively prime which is the case when the left hand side is defined b 1 mod n b mod n mod n 1 distributive a b mod n a mod n b mod n mod n ab mod n a mod n b mod n mod n division definition a b mod n a mod n b 1 mod n mod n when the right hand side is defined that is when b and n are coprime and undefined otherwise inverse multiplication ab mod n b 1 mod n mod n a mod n in programming languages edit modulo operators in various programming languages language operator integer floating point definition abap mod yes yes euclidean 8 actionscript yes yes truncated 9 ada mod yes no floored 10 rem yes no truncated 10 algol 68 mod b yes no euclidean 11 ampl mod yes no truncated apl c yes yes floored applescript mod yes yes truncated autolisp rem yes yes truncated awk yes yes truncated same as fmod in c 12 basic mod yes no varies by implementation bc yes yes truncated 13 c c div yes no truncated d fmod c std fmod c no yes truncated 16 remainder c std remainder c no yes rounded 17 c yes yes truncated 18 math ieeeremainder no yes rounded 19 clarion yes no truncated clean rem yes no truncated clojure mod yes no floored 20 rem yes no truncated 21 cobol function mod yes no floored 22 function rem yes yes truncated 22 coffeescript yes no truncated same as javascript 23 yes no floored 23 coldfusion mod yes no truncated common intermediate language rem signed yes yes truncated 24 rem un unsigned yes no n a common lisp mod yes yes floored 25 rem yes yes truncated 25 crystal modulo yes yes floored remainder yes yes truncated css mod yes yes floored 26 rem yes yes truncated 27 d yes yes truncated 28 dart yes yes euclidean 29 remainder yes yes truncated 30 dc yes yes truncated 31 eiffel yes no truncated elixir rem 2 yes no truncated 32 integer mod 2 yes no floored 33 elm modby yes no floored 34 remainderby yes no truncated 35 erlang rem yes no truncated math fmod 2 no yes truncated same as c 36 euphoria remainder yes yes truncated 37 mod yes yes floored 38 f yes yes truncated same as c 39 math ieeeremainder no yes rounded 19 factor mod yes yes truncated 40 rem yes yes euclidean 41 filemaker mod yes no floored forth mod yes no implementation defined fm mod yes no floored sm rem yes no truncated fortran mod yes yes truncated 42 modulo yes yes floored 43 frink mod yes yes floored 44 full basic mod yes yes floored 45 remainder yes yes truncated 46 glsl yes no undefined 47 mod no yes floored 48 gamemaker studio gml mod yes no truncated gdscript godot yes no truncated 49 posmod yes no euclidean 50 fmod no yes truncated 51 fposmod no yes euclidean 52 go yes no truncated 53 math mod no yes truncated 54 big int mod yes no euclidean 55 big int rem yes no truncated 56 groovy yes no truncated haskell mod yes no floored 57 rem yes no truncated 57 data fixed mod ghc no yes floored haxe yes yes truncated 58 hlsl yes yes undefined 59 j c yes no floored java yes yes truncated 60 integer remainderunsigned long remainderunsigned yes no n a 61 62 math ceilmod strictmath ceilmod yes no ceiling 63 64 65 66 67 68 math floormod strictmath floormod yes no floored 69 70 71 72 73 74 math ieeeremainder strictmath ieeeremainder no yes rounded 75 76 a mod b biginteger yes no euclidean 77 a remainder b biginteger yes no truncated 78 a remainder b a remainder b mathcontext bigdecimal no yes truncated 79 80 javascript typescript yes yes truncated julia mod yes yes floored 81 rem yes yes truncated 82 kotlin rem yes yes truncated 83 mod yes yes floored 84 ksh yes no truncated same as posix sh fmod no yes truncated labview mod yes yes truncated libreoffice mod yes yes floored 85 logo modulo yes no floored remainder yes no truncated lua 5 yes yes floored lua 4 mod x y yes yes truncated liberty basic mod yes no truncated mathcad mod x y yes no floored maple e mod m by default modp e m yes no euclidean mods e m yes no rounded frem e m yes yes rounded mathematica mod a b yes yes floored 86 matlab mod yes yes floored 87 rem yes yes truncated 88 maxima mod yes no floored remainder yes no truncated maya embedded language yes no truncated fmod no yes truncated same as c 89 microsoft excel mod yes yes floored 90 minitab mod yes no floored modula 2 mod yes no floored rem yes no truncated mumps yes no floored 91 netwide assembler nasm nasmx div unsigned yes no n a signed yes no implementation defined 92 nim mod yes yes truncated 93 94 floormod yes yes floored 95 oberon mod yes no floored like e objective c yes no truncated same as c99 object pascal delphi mod yes no truncated ocaml mod yes no truncated 96 mod_float no yes truncated 97 occam yes no truncated pascal iso 7185 and 10206 mod yes no euclidean like f perl yes no floored g posix fmod no yes truncated same as c 99 php yes no truncated 100 fmod no yes truncated 101 pic basic pro yes no truncated pl i mod yes no floored ansi pl i powershell yes no truncated programming code prc math op mod yes no undefined progress modulo yes no truncated prolog mod yes no floored rem yes no truncated purebasic yes no truncated mod x y no yes truncated 102 purescript mod yes no euclidean 103 pure data yes no truncated same as c mod yes no floored python yes yes floored 104 math fmod no yes truncated 105 math remainder no yes rounded 106 q yes no truncated 107 r yes yes floored 108 racket modulo yes some h floored 110 remainder yes some h truncated 111 raku no yes floored realbasic mod yes no truncated reason mod yes no truncated rexx yes yes truncated rpg rem yes no truncated ruby modulo yes yes floored remainder yes yes truncated rust yes yes truncated 112 rem_euclid yes yes euclidean 113 sas mod yes no truncated scala yes yes truncated scheme modulo yes no floored remainder yes no truncated scheme r 6 rs mod yes no euclidean 114 mo...
|
