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Commutative property
Commutative property
Commutative property
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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.

Key Information

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 on a set S is commutative if for all .[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 if[4]

So, an operation is commutative if every two elements commute.[4] An operation is noncommutative if there are two elements such that 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

Noncommutative operations

[edit]
  • Division is noncommutative, since . Subtraction is noncommutative, since . However it is classified more precisely as anti-commutative, since for every ⁠⁠ and ⁠⁠. Exponentiation is noncommutative, since (see Equation xy = yx).[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 and . Then and
  • Matrix multiplication of square matrices of a given dimension is a noncommutative operation, except for ⁠⁠ matrices. For example:[12]
  • The vector product (or cross product) of two vectors in three dimensions is anti-commutative; i.e., .[13]

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,

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]

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 known use of the term was in a French Journal published in 1814

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.

Notes

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Commutative property

View on Grokipedia
from Grokipedia
In , the commutative property is a fundamental stating that, for a * on a set S, two elements x and y satisfy x * y = y * x, meaning the order of the operands does not affect the result. This property holds for and of real numbers, where a + b = b + a and a × b = b × a for any real numbers a and b. It does not apply to operations like or division, where changing the order alters the outcome, such as 5 - 3 ≠ 3 - 5. The commutative property underpins many algebraic manipulations and simplifications, allowing rearrangement of terms in expressions without changing their value, which is essential in solving equations and performing arithmetic efficiently. For instance, in addition, 2 + 7 = 7 + 2 = 9, and in multiplication, 4 × 6 = 6 × 4 = 24, demonstrating its consistency across basic operations on whole numbers, integers, rationals, and reals. This property facilitates mental math strategies and is one of the core axioms defining the structure of number systems like the real numbers. In , commutativity extends beyond basic arithmetic to define structures such as , where is commutative alongside , forming the basis for advanced topics in and . For example, the integers under and form a commutative ring, enabling the study of ideals and polynomials. While not all operations or structures are commutative—such as , where AB ≠ BA in general—the property remains a key distinction in classifying algebraic systems.

Fundamentals

Definition

The commutative property is a fundamental characteristic of certain binary operations in mathematics, where a binary operation is defined as a function that takes two elements from a given set and produces a single element within that same set as output. This property specifically states that interchanging the order of the two input elements, or operands, does not alter the result of the operation. In everyday arithmetic, commutativity manifests intuitively through operations like and , allowing users to rearrange numbers freely to streamline computations—for instance, calculating totals or products without concern for sequence. This flexibility simplifies mental math and problem-solving in practical scenarios, such as tallying quantities or scaling measurements, by emphasizing the operation's over rigid ordering. While the commutative property is a universal feature in familiar school-level , it does not hold for all operations in more advanced mathematical contexts, where order can significantly impact outcomes.

Formal Statement

The commutative property is formally defined for a on a set. Specifically, let SS be a set and ∗:S×S→S* : S \times S \to S a on SS. The operation ∗* is commutative if ∀a,b∈S,a∗b=b∗a.\forall a, b \in S, \quad a * b = b * a. The universal quantifier ∀\forall in this statement indicates that the equality must hold universally for every of elements (a,b)(a, b) with a,b∈Sa, b \in S, without exception, thereby applying the property exhaustively to the entire set SS. This equality a∗b=b∗aa * b = b * a asserts that the output of the operation, which is an element of SS, is identical regardless of the order in which the operands aa and bb are applied, reflecting a in the operation's behavior. The property may fail to hold if the operation is not a on SS, for instance, when SS is not closed under ∗* (meaning a∗b∉Sa * b \notin S for some a,b∈Sa, b \in S), or if, despite closure, there exist a,b∈Sa, b \in S such that a∗b≠b∗aa * b \neq b * a.

Examples

Commutative Operations

The commutative property is exemplified by the of real numbers, where for all a,b∈Ra, b \in \mathbb{R}, a+b=b+aa + b = b + a. This equality is a fundamental in the field structure of the real numbers, ensuring that the order of summands does not affect the result. A proof sketch relies on the axiomatic : the real numbers form an where satisfies commutativity directly as part of the field axioms (A2), alongside associativity and the existence of identities and inverses; in constructive realizations like Dedekind cuts, commutativity emerges from the corresponding property in the rational numbers, where cuts α+β={r+s∣r∈α,s∈β}\alpha + \beta = \{r + s \mid r \in \alpha, s \in \beta\} and β+α={s+r∣s∈β,r∈α}\beta + \alpha = \{s + r \mid s \in \beta, r \in \alpha\} coincide since rational commutes. Simple algebraic verification confirms this: consider (a+b)−(b+a)(a + b) - (b + a); by the field's and cancellation laws, this difference is zero, affirming equality. Multiplication of real numbers also obeys the commutative property: for all a,b∈Ra, b \in \mathbb{R}, a×b=b×aa \times b = b \times a. This is another field axiom (M2), applicable universally, including cases involving zero where a×0=0×a=0a \times 0 = 0 \times a = 0 (derived from the multiplicative identity and axioms) and negative numbers, as multiplication distributes over addition and inherits commutativity from the field's structure. Verification through manipulation shows (a×b)−(b×a)=0(a \times b) - (b \times a) = 0, leveraging the field's properties without contradiction. For instance, with negatives, (−3)×4=4×(−3)=−12(-3) \times 4 = 4 \times (-3) = -12, consistent with the sign rules embedded in the axioms. In Rn\mathbb{R}^n, vector addition is commutative, defined component-wise: for vectors u=(u1,…,un)\mathbf{u} = (u_1, \dots, u_n) and v=(v1,…,vn)\mathbf{v} = (v_1, \dots, v_n), u+v=(u1+v1,…,un+vn)=(v1+u1,…,vn+un)=v+u\mathbf{u} + \mathbf{v} = (u_1 + v_1, \dots, u_n + v_n) = (v_1 + u_1, \dots, v_n + u_n) = \mathbf{v} + \mathbf{u}, since real addition commutes in each coordinate./03%3A_Vector_Spaces_and_Metric_Spaces/3.05%3A_Vector_Spaces._The_Space_C._Euclidean_Spaces) This holds for any dimension, simplifying computations like formations where order-independent sums yield the same resultant vector. A related instance arises in exponentiation with positive integer exponents and the same nonzero base: for a∈R∖{0}a \in \mathbb{R} \setminus \{0\} and positive integers m,nm, n, am⋅an=an⋅am=am+na^m \cdot a^n = a^n \cdot a^m = a^{m+n}, where the multiplication of powers commutes due to the underlying commutative multiplication in the repeated factors defining the exponents. This property is limited to positive integer exponents to avoid issues with fractional or negative powers, which may not preserve the structure in all cases, but verifies through expansion: both sides equal a⋅a⋯aa \cdot a \cdots a (m+nm+n times), reordered freely by multiplication's commutativity.

Noncommutative Operations

Subtraction on the set of real numbers is a binary operation that is not commutative, meaning that for real numbers aa and bb, a−b≠b−aa - b \neq b - a in general. For instance, 3−1=23 - 1 = 2, but 1−3=−21 - 3 = -2. Division on the set of nonzero real numbers is another binary operation that fails to be commutative, as a/b≠b/aa / b \neq b / a except in special cases such as when a=ba = b. An example is 6/2=36 / 2 = 3, whereas 2/6=1/32 / 6 = 1/3.- https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Online_Dictionary_of_Crystallography_(IUCr_Commission)/01%3A_Fundamental_Crystallography/1.10%3A_Binary_Operation Matrix multiplication provides a classic example of noncommutativity in linear . For square matrices over the real numbers, the product ABAB does not generally equal BABA. Consider the 2×2 matrices A=(1101),B=(1011).A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}. Computing the products yields AB=(1101)(1011)=(2111),AB = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}, BA=(1011)(1101)=(1112).BA = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}. Since AB≠BAAB \neq BA, matrix multiplication is not commutative./02%3A_Matrix_Arithmetic/2.02%3A_Matrix_Multiplication) Function composition, denoted by ∘\circ, is also noncommutative as an operation on functions. For functions ff and gg, (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x)) typically differs from (g∘f)(x)=g(f(x))(g \circ f)(x) = g(f(x)). Take f(x)=x+1f(x) = x + 1 and g(x)=2xg(x) = 2x; then (f∘g)(x)=2x+1(f \circ g)(x) = 2x + 1, but (g∘f)(x)=2(x+1)=2x+2(g \circ f)(x) = 2(x + 1) = 2x + 2. The noncommutativity of these operations highlights the importance of order in sequences of applications, such as successive transformations in or computations in algorithms, where reversing the order can yield entirely different results—contrasting with commutative operations like ./02%3A_Introduction_to_Groups/2.02%3A_Binary_Operation)

Commutative Algebraic Structures

Rings

In , a ring is a set RR equipped with two binary operations, and multiplication, such that (R,+)(R, +) forms an , multiplication is associative, and multiplication distributes over addition on both sides. Specifically, for all a,b,c∈Ra, b, c \in R, the distributive laws hold: a⋅(b+c)=a⋅b+a⋅ca \cdot (b + c) = a \cdot b + a \cdot c and (a+b)⋅c=a⋅c+b⋅c(a + b) \cdot c = a \cdot c + b \cdot c. A is a ring where additionally satisfies the commutative property: for all a,b∈Ra, b \in R, a⋅b=b⋅aa \cdot b = b \cdot a. This commutativity simplifies many structural properties, such as the behavior of ideals, which are subsets closed under and under by any ring element. In commutative rings, all ideals are two-sided by default, unlike in noncommutative settings. The integers [Z](/page/Z)\mathbb{[Z](/page/Z)} form a commutative ring under the standard and operations, with 0 as the and every integer having an . Similarly, the ring of polynomials R\mathbb{R} over numbers is a , where and are defined polynomial-wise, and the zero polynomial serves as the . Units in a are elements u∈Ru \in R such that there exists v∈Rv \in R with u⋅v=1u \cdot v = 1, assuming the ring has a multiplicative identity 1; for instance, ±1\pm 1 are the units in Z\mathbb{Z}. Every field is a commutative ring, since fields require commutative multiplication along with every nonzero element having a multiplicative inverse, but the converse does not hold—for example, Z\mathbb{Z} is a commutative ring without being a field, as most elements lack multiplicative inverses within the ring. Commutative ring theory plays a foundational role in modern algebra, underpinning areas like algebraic geometry and number theory through the study of ideals and modules.

Groups and Monoids

A monoid consists of a set MM equipped with a binary operation ⋅:M×M→M\cdot: M \times M \to M that is associative, meaning (a⋅b)⋅c=a⋅(b⋅c)(a \cdot b) \cdot c = a \cdot (b \cdot c) for all a,b,c∈Ma, b, c \in M, and an identity element e∈Me \in M such that a⋅e=e⋅a=aa \cdot e = e \cdot a = a for all a∈Ma \in M. A commutative monoid is a monoid in which the operation is also commutative, satisfying a⋅b=b⋅aa \cdot b = b \cdot a for all a,b∈Ma, b \in M. A group extends the structure of a by requiring that every element a∈Ga \in G has a two-sided inverse a−1∈Ga^{-1} \in G such that a⋅a−1=a−1⋅a=ea \cdot a^{-1} = a^{-1} \cdot a = e. An abelian group, also known as a commutative group, is a group (G,⋅)(G, \cdot) where the operation satisfies commutativity for all elements: ∀a,b∈G,a⋅b=b⋅a.\forall a, b \in G, \quad a \cdot b = b \cdot a. This property distinguishes abelian groups from non-abelian ones, where the order of operation matters. Classic examples illustrate these concepts. The additive group of integers (Z,+)(\mathbb{Z}, +) forms an abelian group, with identity 0 and inverses given by negation, since addition commutes: m+n=n+mm + n = n + m for all m,n∈Zm, n \in \mathbb{Z}. In contrast, the general linear group GL(n,R)\mathrm{GL}(n, \mathbb{R}), consisting of all invertible n×nn \times n matrices over the reals under matrix multiplication, is non-abelian for n≥2n \geq 2, as matrix multiplication generally does not commute. A key feature in group theory is the [G,G][G, G], defined as the generated by all [a,b]=aba−1b−1[a, b] = a b a^{-1} b^{-1} for a,b∈Ga, b \in G. In an , every equals the identity, so [G,G]={e}[G, G] = \{e\}, rendering the group abelian if and only if its is trivial. This trivial simplifies the study of symmetries in , particularly in , where all irreducible representations are one-dimensional, facilitating analysis of group actions on vector spaces.

Historical Development

Etymology

The term "commutative" derives from the Latin commūtātīvus, an adjective formed from commūtāre, meaning "to exchange" or "to change mutually," emphasizing the idea of interchangeability between elements. This Latin root entered mathematical terminology through French, where commuter means "to switch" or "to substitute," combined with the -ative, which denotes a tendency or relational quality. The word's adoption into English mathematical discourse occurred in the , with the earliest recorded use of "commutative law" appearing in in Duncan F. Gregory's Examples of the Processes of the Differential and Calculus. The first mathematical application of the term "commutative" is attributed to François-Joseph Servois in his 1814 memoir Essai sur une nouvelle méthode d'exposition des principes du calcul différentiel, where he described properties of operators, including those of and , as commutatives. In contrast to related terms like "associative," which stems from the Latin associātus meaning "joined together," "commutative" specifically highlights the mutual exchange without altering the result.

Early Uses and Formalization

In , circa 2000 BCE, mathematicians implicitly employed the commutative property during operations, particularly in practical calculations for areas and volumes as documented in texts like the . This approach allowed factors to be rearranged freely—such as treating length and width interchangeably in rectangular area computations—without altering the resulting product, streamlining computations in a base-10 system augmented by doubling and halving techniques. Greek mathematics advanced this implicit understanding, with Euclid's Elements (circa 300 BCE) assuming commutativity in the context of geometric proportions and , though without explicit declaration as a standalone . In Book VII, Proposition 16, Euclid demonstrates the interchangeability of multiplicand and multiplier using the theory of proportions, effectively proving that the product remains unchanged regardless of order, a result derived from additive commutativity and ratio equivalences like A:B::C:DA : B :: C : D. This geometric framing integrated the property into proportional reasoning, influencing subsequent Euclidean derivations. The marked the formalization of the commutative property amid the rise of . Mathematicians like and developed rigorous treatments of fields and ideals, assuming commutativity in these ring-like structures. contributed to this evolution through his explorations of algebraic systems, including commutative cases in his early work on linear algebra and dynamics, though his later quaternions highlighted noncommutative alternatives, prompting deeper scrutiny of the property's scope. These efforts shifted focus from concrete arithmetic to abstract axiomatic frameworks, laying groundwork for modern algebra. Around 1900, advanced the role of commutativity in through his foundational work on and the basis theorem, which established finite generation for ideals in polynomial rings over fields—structures inherently commutative—emphasizing their utility in and . This period solidified commutative rings as a central object of study, bridging 19th-century developments with emerging abstract methods. Post-1930s, the commutative property gained a universal axiomatic footing in the framework of and , particularly through Garrett Birkhoff's 1935 treatise On the Structure of Abstract Algebras, which axiomatized algebraic varieties and incorporated commutativity as a key relational property across diverse structures like groups and rings. This integration within Zermelo-Fraenkel ensured the property's consistency in foundational mathematics, enabling its application in broader categorical and logical contexts without reliance on specific numerical interpretations.

References

  1. https://mathworld.wolfram.com/Commutative.html
  2. https://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut8_property.htm
  3. https://libanswers.centralgatech.edu/AcademicSupport/faq/398648
  4. https://www.math.stonybrook.edu/Videos/MAP103Online/Handouts/Lecture-04-Handout.pdf
  5. https://www.tsc.fl.edu/media/divisions/learning-commons/resources-by-subject/math/foundational-math/number-sense/Properties-of-Real-Numbers.pdf
  6. https://mathworld.wolfram.com/CommutativeAlgebra.html
  7. https://mathworld.wolfram.com/Ring.html
  8. https://mathworld.wolfram.com/CommutingMatrices.html
  9. https://math.berkeley.edu/~arash/113/notes/2_3.pdf
  10. https://open.maricopa.edu/mathforelementaryteachers/chapter/properties-of-operations/
  11. https://archive.cbts.edu/Textbook/63xGs8/897090/Definition_Of_Commutative_Property_In_Math.pdf
  12. https://faculty.etsu.edu/gardnerr/4127/notes/I-2.pdf
  13. https://faculty.etsu.edu/gardnerr/4127/notes/Glossary.pdf
  14. https://www.cs.cmu.edu/afs/cs/academic/class/15251-f07/Site/Materials/Lectures/Lecture17/lecture17.pdf
  15. https://sites.math.washington.edu/~mathcircle/circle/archive/2013-14/advanced/mc-13a-w14.pdf
  16. https://sites.math.washington.edu/~hart/m524/realprop.pdf
  17. https://math-sites.uncg.edu/sites/pauli/112/HTML/secexp.html
  18. https://math.colgate.edu/math320/dlantz/extras/notes01.pdf
  19. https://math.hawaii.edu/~tom/old_classes/412notes3.pdf
  20. https://people.reed.edu/~jerry/332/16ideals.pdf
  21. https://kconrad.math.uconn.edu/blurbs/ringtheory/ideals.pdf
  22. https://www-users.cse.umn.edu/~garrett/m/algebra/notes_2023-24/04.pdf
  23. https://sites.millersville.edu/bikenaga/linear-algebra/rings/rings.pdf
  24. https://math.ou.edu/~cremling/teaching/lecturenotes/alg/ln2.pdf
  25. https://www.math.lsu.edu/~madden/M7290Fall2006/lecture3.pdf
  26. https://faculty.etsu.edu/gardnerr/5410/notes/I-1.pdf
  27. https://www.math.purdue.edu/~arapura/algebra/ch3.pdf
  28. https://math.okstate.edu/people/binegar/3613/3613-l21.pdf
  29. https://www.math.purdue.edu/~arapura/algebra/ch15
  30. https://www.math.wm.edu/~vinroot/430S11Commutators.pdf
  31. https://chaosbk.physics.gatech.edu/library/ArWeHa13chap17Group-Theory.pdf
  32. https://www.etymonline.com/word/commutative
  33. https://en.wiktionary.org/wiki/commutative
  34. https://mathshistory.st-andrews.ac.uk/Miller/mathword/c/
  35. https://mathshistory.st-andrews.ac.uk/Biographies/Servois/
  36. https://brill.com/display/book/9789004691568/BP000020.xml
  37. https://polytropy.com/2017/01/02/the-geometry-of-numbers-in-euclid/
  38. https://mathshistory.st-andrews.ac.uk/HistTopics/Ring_theory/
  39. https://www.gresham.ac.uk/watch-now/hamilton-boole-and-their-algebras
  40. https://math.hawaii.edu/~ralph/Classes/619/univ-algebra.pdf
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