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commutative property wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 definition 2 examples toggle examples subsection 2 1 commutative operations 2 2 noncommutative operations 3 commutative structures 4 history and etymology 5 see also 6 notes 7 references toggle the table of contents commutative property 71 languages afrikaans العربية asturianu azərbaycanca башҡортса беларуская български বাংলা bosanski català کوردی čeština чӑвашла dansk deutsch ελληνικά esperanto español eesti euskara فارسی suomi français nordfriisk gaeilge gàidhlig galego עברית हिन्दी hrvatski magyar հայերեն interlingua bahasa indonesia íslenska italiano 日本語 қазақша 한국어 latina lietuvių latviešu македонски മലയാളം bahasa melayu မြန်မာဘာသာ nederlands norsk nynorsk norsk bokmål polski português română русский srpskohrvatski српскохрватски සිංහල simple english slovenčina slovenščina shqip српски srpski svenska தமிழ் ไทย türkçe татарча tatarça українська vèneto tiếng việt 吴语 粵語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikimedia commons wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia redirected from commutativity property of some mathematical operations commutative redirects here for other uses see commutative disambiguation commutative property type property field algebra statement a binary operation is commutative if changing the order of the operands does not change the result symbolic statement x y y x x y s displaystyle x y y x quad forall x y in s in mathematics a binary operation is commutative if changing the order of the operands does not change the result it is a fundamental property of many binary operations and many mathematical proofs depend on it perhaps most familiar as a property of arithmetic e g 3 4 4 3 or 2 5 5 2 the property can also be used in more advanced settings the name is needed because there are operations such as division and subtraction that do not have it for example 3 5 5 3 such operations are not commutative and so are referred to as noncommutative operations the idea that simple operations such as the multiplication and addition of numbers are commutative was for many centuries implicitly assumed thus this property was not named until the 19th century when new algebraic structures started to be studied 1 definition edit a binary operation displaystyle on a set s is commutative if x y y x displaystyle x y y x for all x y s displaystyle x y in s 2 an operation that is not commutative is said to be noncommutative 3 one says that x commutes with y or that x and y commute under displaystyle if 4 x y y x displaystyle x y y x so an operation is commutative if every two elements commute 4 an operation is noncommutative if there are two elements such that x y y x displaystyle x y neq y x this does not exclude the possibility that some pairs of elements commute 3 examples edit the cumulation of apples which can be seen as an addition of natural numbers is commutative commutative operations edit the addition of vectors is commutative because a b b a displaystyle vec a vec b vec b vec a addition and multiplication are commutative in most number systems and in particular between natural numbers integers rational numbers real numbers and complex numbers this is also true in every field 5 addition is commutative in every vector space and in every algebra 6 union and intersection are commutative operations on sets 7 and and or are commutative logical operations 8 noncommutative operations edit division is noncommutative since 1 2 2 1 displaystyle 1 div 2 neq 2 div 1 subtraction is noncommutative since 0 1 1 0 displaystyle 0 1 neq 1 0 however it is classified more precisely as anti commutative since x y y x displaystyle x y y x for every x displaystyle x and y displaystyle y exponentiation is noncommutative since 2 3 3 2 displaystyle 2 3 neq 3 2 see equation x y y x 9 some truth functions are noncommutative since their truth tables are different when one changes the order of the operands 10 for example the truth tables for a b a b and b a a b are a b a b b a f f t t f t t f t f f t t t t t function composition is generally noncommutative 11 for example if f x 2 x 1 displaystyle f x 2x 1 and g x 3 x 7 displaystyle g x 3x 7 then f g x f g x 2 3 x 7 1 6 x 15 displaystyle f circ g x f g x 2 3x 7 1 6x 15 and g f x g f x 3 2 x 1 7 6 x 10 displaystyle g circ f x g f x 3 2x 1 7 6x 10 matrix multiplication of square matrices of a given dimension is a noncommutative operation except for 1 1 displaystyle 1 times 1 matrices for example 12 0 2 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 displaystyle begin bmatrix 0 2 0 1 end bmatrix begin bmatrix 1 1 0 1 end bmatrix begin bmatrix 0 1 0 1 end bmatrix neq begin bmatrix 0 1 0 1 end bmatrix begin bmatrix 1 1 0 1 end bmatrix begin bmatrix 0 1 0 1 end bmatrix the vector product or cross product of two vectors in three dimensions is anti commutative i e b a a b displaystyle mathbf b times mathbf a mathbf a times mathbf b 13 the quaternion multiplication is noncommutative since i j j i displaystyle mathbf ij mathbf ji where i displaystyle mathbf i and j displaystyle mathbf j are two of the basis quaternions commutative structures edit some types of algebraic structures involve an operation that does not require commutativity if this operation is commutative for a specific structure the structure is often said to be commutative so a commutative semigroup is a semigroup whose operation is commutative 14 a commutative monoid is a monoid whose operation is commutative 15 a commutative group or abelian group is a group whose operation is commutative 16 a commutative ring is a ring whose multiplication is commutative addition in a ring is always commutative 17 however in the case of algebras the phrase commutative algebra refers only to associative algebras that have a commutative multiplication 18 history and etymology edit the first known use of the term was in a french journal published in 1814 records of the implicit use of the commutative property go back to ancient times the egyptians used the commutative property of multiplication to simplify computing products 19 euclid is known to have assumed the commutative property of multiplication in his book elements 20 formal uses of the commutative property arose in the late 18th and early 19th centuries when mathematicians began to work on a theory of functions nowadays the commutative property is a well known and basic property used in most branches of mathematics 2 the first recorded use of the term commutative was in a memoir by françois servois in 1814 which used the word commutatives when describing functions that have what is now called the commutative property 21 commutative is the feminine form of the french adjective commutatif which is derived from the french noun commutation and the french verb commuter meaning to exchange or to switch a cognate of to commute the term then appeared in english in 1838 in duncan gregory s article entitled on the real nature of symbolical algebra published in 1840 in the transactions of the royal society of edinburgh 22 see also edit look up commutative property in wiktionary the free dictionary anticommutative property canonical commutation relation in quantum mechanics centralizer and normalizer also called a commutant commutative diagram commutative neurophysiology commutator commuting graph commuting probability particle statistics for commutativity in physics quasi commutative property symmetric function a binary operation is commutative if and only if it is a symmetric function of its two arguments trace monoid notes edit rice 2011 p 4 1 2 saracino 2008 p 11 1 2 hall 1966 pp 262 263 1 2 lovett 2022 p 12 rosen 2013 see the appendix i sterling 2009 p 248 johnson 2003 p 642 o regan 2008 p 33 posamentier et al 2013 p 71 medina et al 2004 p 617 tarasov 2008 p 56 cooke 2014 p 7 haghighi kumar mishev 2024 p 118 grillet 2001 pp 1 2 grillet 2001 p 3 gallian 2006 p 34 gallian 2006 p 236 tuset 2025 p 99 gay shute 1987 p 16 17 barbeau 1968 p 183 see book vii proposition 5 in david e joyce s online edition of euclid s elements allaire bradley 2002 rice 2011 p 4 gregory 1840 references edit allaire patricia r bradley robert e 2002 symbolical algebra as a foundation for calculus d f gregory s contribution historia mathematica 29 4 395 426 doi 10 1006 hmat 2002 2358 barbeau alice mae 1968 a historical approach to the theory of groups vol 2 university of wisconsin madison cooke richard g 2014 infinite matrices and sequence spaces dover publications isbn 978 0 486 78083 2 gallian joseph 2006 contemporary abstract algebra 6e ed houghton mifflin isbn 0 618 51471 6 gay robins r shute charles c d 1987 the rhind mathematical papyrus an ancient egyptian text british museum isbn 0 7141 0944 4 gregory d f 1840 on the real nature of symbolical algebra transactions of the royal society of edinburgh 14 208 216 grillet p a 2001 commutative semigroups advances in mathematics vol 2 dordrecht kluwer academic publishers doi 10 1007 978 1 4757 3389 1 isbn 0 7923 7067 8 mr 2017849 haghighi aliakbar montazer kumar abburi anil mishev dimitar 2024 higher mathematics for science and engineering springer isbn 978 981 99 5431 5 hall f m 1966 an introduction to abstract algebra volume 1 new york cambridge university press mr 0197233 johnson james l 2003 probability and statistics for computer science john wiley sons isbn 978 0 471 32672 4 lovett stephen 2022 abstract algebra a first course crc press isbn 978 1 000 60544 0 medina jesús ojeda aciego manuel valverde agustín vojtáš peter 2004 towards biresiduated multi adjoint logic programming in conejo ricardo urretavizcaya maite pérez de la cruz josé luis eds current topics in artificial intelligence 10th conference of the spanish association for artificial intelligence caepia 2003 and 5th conference on technology transfer ttia 2003 november 12 14 2003 lecture notes in computer science vol 3040 san sebastian spain springer doi 10 1007 b98369 isbn 978 3 540 22218 7 o regan gerard 2008 a brief history of computing springer isbn 978 1 84800 083 4 posamentier alfred s farber william germain williams terri l paris elaine thaller bernd lehmann ingmar 2013 100 commonly asked questions in math class corwin press isbn 978 1 4522 4308 5 rice adrian 2011 introduction in flood raymond rice adrian wilson robin eds mathematics in victorian britain oxford university press isbn 9780191627941 rosen kenneth 2013 discrete maths and its applications global edition mcgraw hill isbn 978 0 07 131501 2 saracino dan 2008 abstract algebra a first course 2nd ed waveland press inc sterling mary j 2009 linear algebra for dummies john wiley sons isbn 978 0 470 43090 3 tarasov vasily 2008 quantum mechanics of non hamiltonian and dissipative systems vol 7 1st ed elsevier isbn 978 0 08 055971 1 tuset lars 2025 abstract algebra via numbers cham springer doi 10 1007 978 3 031 74623 9 isbn 978 3 031 74622 2 mr 4886847 retrieved from https en wikipedia org w index php title commutative_property oldid 1374440636 categories properties of binary operations elementary algebra rules of inference symmetry concepts in physics functional analysis hidden categories articles with short description short description is different from wikidata use dmy dates from june 2023 good articles this page was last edited on 12 september 2026 at 02 16 utc 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