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Commutative property

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Property of some mathematical operations

"Commutative" redirects here. For other uses, seeCommutative (disambiguation).

Commutative property

Type	Property
Field	Algebra
Statement	Abinary operation iscommutative if changing the order of theoperands does not change the result.
Symbolic statement	$xy=yx\quad \forall x,y\in S.$

Inmathematics, abinary operation iscommutative if changing the order of theoperands does not change the result. It is a fundamental property of many binary operations, and manymathematical 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 asdivision andsubtraction, that do not have it (for example,"3 − 5 ≠ 5 − 3"); such operations arenot commutative, and so are referred to asnoncommutative operations.

The idea that simple operations, such as themultiplication andaddition of numbers, are commutative was for many centuries implicitly assumed. Thus, this property was not named until the 19th century, when newalgebraic structures started to be studied.^[1]

Definition

Abinary operation $*$ on asetS iscommutative if $x*y=y*x$ for all $x,y\in S$ .^[2] An operation that is not commutative is said to benoncommutative.^[3]

One says thatxcommutes withy or thatx andycommute under $*$ if^[4] $x*y=y*x.$

So, an operation is commutative if every two elements commute.^[4] An operation is noncommutative if there are two elements such that $x*y\neq y*x.$ This does not exclude the possibility that some pairs of elements commute.^[3]

Examples

The cumulation of apples, which can be seen as an addition of natural numbers, is commutative.

Commutative operations

The addition of vectors is commutative, because ${\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}.$

{\displaystyle {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}.} — The addition of vectors is commutative, because ${\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}.$

Addition andmultiplication are commutative in mostnumber systems, and, in particular, betweennatural numbers,integers,rational numbers,real numbers andcomplex numbers. This is also true in everyfield.^[5]
Addition is commutative in everyvector space and in everyalgebra.^[6]
Union andintersection are commutative operations onsets.^[7]
"And" and "or" are commutativelogical operations.^[8]

Noncommutative operations

Division is noncommutative, since $1\div 2\neq 2\div 1$ .Subtraction is noncommutative, since $0-1\neq 1-0$ . However it is classified more precisely asanti-commutative, since $x-y=-(y-x)$ for every⁠ $x {\displaystyle x}$ ⁠ and⁠ $y {\displaystyle y}$ ⁠.Exponentiation is noncommutative, since $2^{3}\neq 3^{2}$ (seeEquationx^y =y^x.^[9]
Sometruth functions are noncommutative, since theirtruth 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 $f(x)=2x+1$ and $g(x)=3x+7$ . Then $(f\circ g)(x)=f(g(x))=2(3x+7)+1=6x+15$ and $(g\circ f)(x)=g(f(x))=3(2x+1)+7=6x+10.$
Matrix multiplication ofsquare matrices of a given dimension is a noncommutative operation, except for⁠ $1\times 1$ ⁠ matrices. For example:^[12] ${\begin{bmatrix}0&2\\0&1\end{bmatrix}}={\begin{bmatrix}1&1\\0&1\end{bmatrix}}{\begin{bmatrix}0&1\\0&1\end{bmatrix}}\neq {\begin{bmatrix}0&1\\0&1\end{bmatrix}}{\begin{bmatrix}1&1\\0&1\end{bmatrix}}={\begin{bmatrix}0&1\\0&1\end{bmatrix}}$
The vector product (orcross product) of two vectors in three dimensions isanti-commutative; i.e., $\mathbf {b} \times \mathbf {a} =-(\mathbf {a} \times \mathbf {b} )$ .^[13]

Commutative structures

Some types ofalgebraic structures involve an operation that does not require commutativity. If this operation is commutative for a specific structure, the structure is often said to becommutative. So,

acommutative semigroup is asemigroup whose operation is commutative;^[14]
acommutative monoid is amonoid whose operation is commutative;^[15]
acommutative group orabelian group is agroup whose operation is commutative;^[16]
acommutative ring is aring whosemultiplication is commutative. (Addition in a ring is always commutative.)^[17]

However, in the case ofalgebras, the phrase "commutative algebra" refers only toassociative algebras that have a commutative multiplication.^[18]

History and etymology

Records of the implicit use of the commutative property go back to ancient times. TheEgyptians used the commutative property ofmultiplication to simplify computingproducts.^[19]Euclid is known to have assumed the commutative property of multiplication in his bookElements.^[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 known use of the term was in a French Journal published in 1814

The first recorded use of the termcommutative was in a memoir byFrançois Servois in 1814, which used the wordcommutatives when describing functions that have what is now called the commutative property.^[21]Commutative is the feminine form of the French adjectivecommutatif, which is derived from the French nouncommutation and the French verbcommuter, meaning "to exchange" or "to switch", a cognate ofto commute. The term then appeared in English in 1838. inDuncan Gregory's article entitled "On the real nature of symbolical algebra" published in 1840 in theTransactions of the Royal Society of Edinburgh.^[22]

See also

Look upcommutative 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
Particle statistics (for commutativity inphysics)
Quasi-commutative property
Trace monoid
Commuting probability

Notes

^Rice 2011, p. 4.
^^a ^bSaracino 2008, p. 11.
^^a ^bHall 1966, pp. 262–263.
^^a ^bLovett 2022, p. 12.
^Rosen 2013, See theAppendix 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. SeeBook VII, Proposition 5, inDavid E. Joyce's online edition of Euclid'sElements
^Allaire & Bradley 2002.
^Rice 2011, p. 4;Gregory 1840.

References

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.

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