# Commutative property
The commutative property is the property of a binary operation that swapping the order of its two inputs never changes the result: for a binary operation \( \bullet \) on a set \( S \), the operation is commutative if \( a \bullet b = b \bullet a \) for all \( a \in S \) and \( b \in S \)[1](https://mat112.uncg.edu/HTML/secbinopcom.html). Addition and multiplication of numbers are the familiar examples; subtraction and division are not commutative[1](https://mat112.uncg.edu/HTML/secbinopcom.html) • [2](https://openstax.org/books/intermediate-algebra-2e/pages/1-5-properties-of-real-numbers).
| Key fact | Detail |
|---|---|
| Definition | A binary operation \( \bullet : S \times S \to S \) is commutative if \( a \bullet b = b \bullet a \) for all \( a, b \in S \)[1](https://mat112.uncg.edu/HTML/secbinopcom.html) |
| Commutative operations | Addition of integers and multiplication of natural numbers (and of real numbers) are commutative[1](https://mat112.uncg.edu/HTML/secbinopcom.html) • [2](https://openstax.org/books/intermediate-algebra-2e/pages/1-5-properties-of-real-numbers) |
| Non-commutative operations | Subtraction (\( 5-2=3 \) but \( 2-5=-3 \)) and division (\( 12 \div 3 \neq 3 \div 12 \)) are not commutative[1](https://mat112.uncg.edu/HTML/secbinopcom.html) • [2](https://openstax.org/books/intermediate-algebra-2e/pages/1-5-properties-of-real-numbers) |
| Matrix multiplication | For suitable square matrices, \( A \times B \neq B \times A \), so matrix multiplication is not commutative[3](https://www.uib.no/en/rg/algebraisk-geometri/57980/6-commutative-versus-noncommutative-systems) |
| Distinct from associativity | Commutativity concerns order of two inputs; associativity concerns grouping of three, \( (xy)z = x(yz) \); an operation can have one property without the other[4](https://math.umd.edu/~pbrosnan/notes/ugalg/sect0008.html) |
| Early record | Euclid's *Elements*, Book VII, Proposition 16 states the commutativity of multiplication of numbers, \( ab = ba \)[5](https://www.euclids-elements.org/elements/books/bookVII/propositions/propVII16/) |
| Floating point | IEEE floating-point addition and multiplication are commutative but not associative[6](https://math.nyu.edu/inmemoriam/donev/Teaching/SciComp-Fall2015/Lecture-IEEE.handout.pdf) |
## Formal definition
The definition is stated for a binary operation, a map \( \bullet : S \times S \to S \) that assigns to each ordered pair of elements of a set \( S \) another element of \( S \)[1](https://mat112.uncg.edu/HTML/secbinopcom.html). The operation is commutative when \( a \bullet b = b \bullet a \) holds for all \( a \in S \) and all \( b \in S \); the quantifier matters, since a single pair with \( a \bullet b \neq b \bullet a \) is enough to show the operation is not commutative[1](https://mat112.uncg.edu/HTML/secbinopcom.html).
In abstract algebra the same property is phrased for a magma, a set equipped with a binary operation: the magma \( M \) is commutative if \( xy = yx \) for all \( x, y \in M \)[4](https://math.umd.edu/~pbrosnan/notes/ugalg/sect0008.html). Notation follows the property: when a commutative operation is written additively as \( (m,n) \mapsto m+n \), the identity element is traditionally denoted \( 0 \); for a non-commutative operation the symbols \( e \) or \( 1 \) are used instead, and the symbol \( + \) is never used for an operation that is not commutative[4](https://math.umd.edu/~pbrosnan/notes/ugalg/sect0008.html).
## Commutative and non-commutative operations on numbers
For the integers, addition and multiplication are commutative: \( 2+7 = 7+2 = 9 \), and in general \( a+b = b+a \) and \( a \cdot b = b \cdot a \) for real numbers \( a \) and \( b \)[1](https://mat112.uncg.edu/HTML/secbinopcom.html) • [2](https://openstax.org/books/intermediate-algebra-2e/pages/1-5-properties-of-real-numbers). Subtraction fails the property, as the counterexample \( 5-2 = 3 \) against \( 2-5 = -3 \) shows, and division fails it as well, since \( 12 \div 3 \neq 3 \div 12 \)[1](https://mat112.uncg.edu/HTML/secbinopcom.html) • [2](https://openstax.org/books/intermediate-algebra-2e/pages/1-5-properties-of-real-numbers).
Commutativity is independent of associativity, the property that \( (xy)z = x(yz) \) for all \( x, y, z \)[4](https://math.umd.edu/~pbrosnan/notes/ugalg/sect0008.html). The Lie bracket on the set of \( 2 \times 2 \) real matrices, defined by \( [A,B] = AB - BA \), is an operation that is neither commutative nor associative; with particular choices of matrices \( A \), \( B \), and \( C \), \( [[A,B],C] \) is the zero matrix while \( [A,[B,C]] \) is not[4](https://math.umd.edu/~pbrosnan/notes/ugalg/sect0008.html).
## Beyond numbers
[Matrix multiplication](https://www.edgepedia.org/matrix-multiplication) is the standard example of a non-commutative operation on familiar objects. For the integers, \( a \times b = b \times a \) and the order of the factors is unimportant; for matrices, computing \( A \times B \) and \( B \times A \) for suitable matrices gives different results, so the multiplication is not commutative[3](https://www.uib.no/en/rg/algebraisk-geometri/57980/6-commutative-versus-noncommutative-systems).
The distinction separates two bodies of mathematical theory. Commutative algebraic systems, which the polynomials constitute, are closely associated with geometric objects, a relationship studied in algebraic geometry, where algebraic and geometric properties reflect one another. Noncommutative systems relate to geometry in a more extrinsic way, often through operations on geometric objects that give symmetries; much of the algebra in modern theoretical physics is noncommutative[3](https://www.uib.no/en/rg/algebraisk-geometri/57980/6-commutative-versus-noncommutative-systems).
## History
The property was recorded explicitly in antiquity. Proposition 16 of Book VII of Euclid's *Elements* states the commutativity of multiplication of formal numbers, \( ab = ba \), and the proposition is used later in the book, including in Proposition VII.18[5](https://www.euclids-elements.org/elements/books/bookVII/propositions/propVII16/). Modern summaries of the *Elements* likewise list VII.16 as the commutativity of multiplication of numbers, \( mn = nm \)[7](https://www.cs.clarku.edu/~djoyce/java/elements/trip.html).
## Commutativity in computation and learning
[Floating-point arithmetic](https://www.edgepedia.org/floating-point-arithmetic) preserves commutativity while losing associativity. IEEE floating-point addition and multiplication are commutative but not associative, because rounding after each operation makes the result depend on the grouping of the computation[6](https://math.nyu.edu/inmemoriam/donev/Teaching/SciComp-Fall2015/Lecture-IEEE.handout.pdf). This is why reordering parenthesized sums of floating-point numbers can change the answer even though reordering two addends cannot.
The property is also a milestone in arithmetic learning. In the Common Core State Standards, grade 3 students apply the commutative property of multiplication as a strategy: knowing \( 6 \times 4 = 24 \) immediately gives \( 4 \times 6 = 24 \)[8](https://www.thecorestandards.org/Math/Content/3/OA/B/5/). Developmental research finds the concept acquired unevenly relative to other additive concepts: among grade 3, 4, and 5 children, understanding of identity and negation was strong, commutativity and inversion were medium, and equivalence and associativity were weak[9](https://www.sciencedirect.com/science/article/pii/S0022096516302442). Classroom studies of kindergarten and first-grade students observed the least sophisticated understandings for the commutative property, with some children showing only instrumental understanding even after instruction[10](https://www.terc.edu/publications/kindergarten-and-first-grade-students-understandings-and-representations-of-arithmetic-properties/).
## References
1. [Section 13.5 Commutativity, MAT 112, University of North Carolina at Greensboro](https://mat112.uncg.edu/HTML/secbinopcom.html)
2. [1.5 Properties of Real Numbers, Intermediate Algebra 2e, OpenStax](https://openstax.org/books/intermediate-algebra-2e/pages/1-5-properties-of-real-numbers)
3. [Commutative versus noncommutative systems, Algebraic Geometry research group, University of Bergen](https://www.uib.no/en/rg/algebraisk-geometri/57980/6-commutative-versus-noncommutative-systems)
4. [UMD 403: Undergraduate Algebra, Binary Operations, University of Maryland](https://math.umd.edu/~pbrosnan/notes/ugalg/sect0008.html)
5. [Euclid's Elements, Book VII, Proposition 16](https://www.euclids-elements.org/elements/books/bookVII/propositions/propVII16/)
6. [Roundoff Errors, IEEE Arithmetic, lecture handout, Courant Institute, New York University](https://math.nyu.edu/inmemoriam/donev/Teaching/SciComp-Fall2015/Lecture-IEEE.handout.pdf)
7. [Euclid's Elements, A Quick Trip, Clark University](https://www.cs.clarku.edu/~djoyce/java/elements/trip.html)
8. [Common Core State Standards, Grade 3, Operations & Algebraic Thinking, standard 3.OA.B.5](https://www.thecorestandards.org/Math/Content/3/OA/B/5/)
9. [Robinson, K. M., Dubé, A. K., Beatch, J.-A. (2017). Children's understanding of additive concepts. Journal of Experimental Child Psychology](https://www.sciencedirect.com/science/article/pii/S0022096516302442)
10. [Kindergarten and First-Grade Students' Understandings and Representations of Arithmetic Properties, TERC](https://www.terc.edu/publications/kindergarten-and-first-grade-students-understandings-and-representations-of-arithmetic-properties/)
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