Commutative property

"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	$x*y=y*x\quad \forall x,y\in S.$

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 years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized.

Definition

A binary operation $*$ on a set S is commutative if $$x*y=y*x$$ for all $x,y\in S$. An operation that is not commutative is said to be noncommutative.

One says that x commutes with y or that x and y commute under $*$ if $$x*y=y*x.$$

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

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 ${\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}.$

• 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.
• Addition is commutative in every vector space and in every algebra.
• Union and intersection are commutative operations on sets.
• "And" and "or" are commutative logical operations.

Noncommutative operations

Division, subtraction, and exponentiation

See also: Equation xy = yx

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 as anti-commutative, since $0-1=-(1-0)$.

Exponentiation is noncommutative, since $2^{3}\neq 3^{2}$. This property leads to two different "inverse" operations of exponentiation (namely, the nth-root operation and the logarithm operation), whereas multiplication only has one inverse operation.

Truth functions

Some truth functions are noncommutative, since the truth tables for the functions are different when one changes the order of the operands. 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 of linear functions

Function composition of linear functions from the real numbers to the real numbers is almost always noncommutative. For example, let $f(x)=2x+1$ and $g(x)=3x+7$. Then $$(f\circ g)(x)=f(g(x))=2(3x+7)+1=6x+15$$ and $$(g\circ f)(x)=g(f(x))=3(2x+1)+7=6x+10$$ This also applies more generally for linear and affine transformations from a vector space to itself.

Matrix multiplication

Matrix multiplication of square matrices is almost always noncommutative, for example: $${\begin{bmatrix}0&2\\0&1\end{bmatrix}}={\begin{bmatrix}1&1\\0&1\end{bmatrix}}{\begin{bmatrix}0&1\\0&1\end{bmatrix}}\neq {\begin{bmatrix}0&1\\0&1\end{bmatrix}}{\begin{bmatrix}1&1\\0&1\end{bmatrix}}={\begin{bmatrix}0&1\\0&1\end{bmatrix}}$$

Vector product

The vector product (or cross product) of two vectors in three dimensions is anti-commutative; i.e., $\mathbf {b} \times \mathbf {a} =-(\mathbf {a} \times \mathbf {b} )$.

Set theory

In group and set theory, many algebraic structures are called commutative when certain operands satisfy the commutative property. In higher branches of mathematics, such as analysis and linear algebra the commutativity of well-known operations (such as addition and multiplication on real and complex numbers) is often used (or implicitly assumed) in proofs.

Mathematical structures and commutativity

• A commutative semigroup is a set endowed with a total, associative and commutative operation.
• If the operation additionally has an identity element, one has a commutative monoid.
• An abelian group, or commutative group is a group whose group operation is commutative.
• A commutative ring is a ring whose multiplication is commutative. (Addition in a ring is always commutative.)
• In a field both addition and multiplication are commutative.

History and etymology

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. Euclid is known to have assumed the commutative property of multiplication in his book Elements. 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.

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. 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.

Non-commuting operators in quantum mechanics

Main article: Canonical commutation relation

In quantum mechanics as formulated by Schrödinger, physical variables are represented by linear operators such as $x$ (meaning multiply by $x$), and ${\textstyle {\frac {d}{dx}}}$. These two operators do not commute as may be seen by considering the effect of their compositions ${\textstyle x{\frac {d}{dx}}}$ and ${\textstyle {\frac {d}{dx}}x}$ (also called products of operators) on a one-dimensional wave function $\psi (x)$: $$x\cdot {\mathrm {d} \over \mathrm {d} x}\psi =x\cdot \psi '\ \neq \ \psi +x\cdot \psi '={\mathrm {d} \over \mathrm {d} x}\left(x\cdot \psi \right)$$

According to the uncertainty principle of Heisenberg, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually complementary, which means they cannot be simultaneously measured or known precisely. For example, the position and the linear momentum in the $x$-direction of a particle are represented by the operators $x$ and $-i\hbar {\frac {\partial }{\partial x}}$, respectively (where $\hbar$ is the reduced Planck constant). This is the same example except for the constant $-i\hbar$, so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary.

See also

• Anticommutative property
• Centralizer and normalizer (also called a commutant)
• Commutative diagram
• Commutative (neurophysiology)
• Commutator
• Parallelogram law
• Particle statistics (for commutativity in physics)
• Proof that Peano's axioms imply the commutativity of the addition of natural numbers
• Quasi-commutative property
• Trace monoid
• Commuting probability

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