A
binary operation is commutative if changing the order of the
operands does not change the result.
Symbolic statement
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.[1][2] A similar property exists for
binary relations; a binary relation is said to be
symmetric if the relation applies regardless of the order of its operands; for example,
equality is symmetric as two equal mathematical objects are equal regardless of their order.[3]
In other words, an operation is commutative if every two elements commute. An operation that does not satisfy the above property is called noncommutative.
One says that xcommutes with y or that x and ycommute under if
That is, a specific pair of elements may commute even if the operation is (strictly) noncommutative.
Exponentiation is noncommutative, since . This property leads to two different "inverse" operations of exponentiation (namely, the
nth-root operation and the
logarithm operation), which is unlike the multiplication. [7]
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
The vector product (or
cross product) of two vectors in three dimensions is
anti-commutative; i.e., b × a = −(a × b).
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.[8][9]Euclid is known to have assumed the commutative property of multiplication in his book
Elements.[10] 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. Today the commutative property is a well-known and basic property used in most branches of mathematics.
The first recorded use of the term commutative was in a memoir by
François Servois in 1814,[1][11] 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.[2] 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.[12]
Commutativity of implication (also called the law of permutation)
Commutativity of equivalence (also called the complete commutative law of equivalence)
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.[16][17][18]
The associative property is closely related to the commutative property. The associative property of an expression containing two or more occurrences of the same operator states that the order operations are performed in does not affect the final result, as long as the order of terms does not change. In contrast, the commutative property states that the order of the terms does not affect the final result.
Most commutative operations encountered in practice are also associative. However, commutativity does not imply associativity. A counterexample is the function
which is clearly commutative (interchanging x and y does not affect the result), but it is not associative (since, for example, but ). More such examples may be found in
commutative non-associative magmas. Furthermore, associativity does not imply commutativity either – for example multiplication of
quaternions or of
matrices is always associative but not always commutative.
Some forms of
symmetry can be directly linked to commutativity. When a commutative operation is written as a
binary function then this function is called a
symmetric function, and its
graph in
three-dimensional space is symmetric across the plane . For example, if the function f is defined as then is a symmetric function.
For relations, a
symmetric relation is analogous to a commutative operation, in that if a relation R is symmetric, then .
In
quantum mechanics as formulated by
Schrödinger, physical variables are represented by
linear operators such as (meaning multiply by ), and . These two operators do not commute as may be seen by considering the effect of their
compositions and (also called products of operators) on a one-dimensional
wave function:
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 -direction of a particle are represented by the operators and , respectively (where is the
reduced Planck constant). This is the same example except for the constant , so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary.