Inmathematics, a binary operationiscommutative 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. Most familiar as the name of the property that says "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 corresponding 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]
Common uses
The commutative property (orcommutative law) is a property generally associated with binary operations and functions. If the commutative property holds for a pair of elements under a certain binary operation then the two elements are said to commute under that operation.
Mathematical definitions
Abinary operation on a setS is called commutativeif[4][5]
An operation that does not satisfy the above property is called non-commutative.
One says that xcommutes with y or that x and ycommute under if
In other words, an operation is commutative if every pair of elements commute.
Putting on socks resembles a commutative operation since which sock is put on first is unimportant. Either way, the result (having both socks on), is the same. In contrast, putting on underwear and trousers is not commutative.
The commutativity of addition is observed when paying for an item with cash. Regardless of the order the bills are handed over in, they always give the same total.
Commutative operations in mathematics
Two well-known examples of commutative binary operations:[4]
For example, 3 × 5 = 5 × 3, since both expressions equal 15.
As a direct consequence of this, it also holds true that expressions on the form y% of z and z% of y are commutative for all real numbers y and z.[6] For example 64% of 50 = 50% of 64, since both expressions equal 32, and 30% of 50% = 50% of 30%, since both of those expressions equal 15%.
Some binary truth functions are also commutative, since the truth tables for the functions are the same when one changes the order of the operands.
For example, the logical biconditional function p ↔ q is equivalent to q ↔ p. This function is also written as p IFF q, or as p ≡ q, or as Epq.
The last form is an example of the most concise notation in the article on truth functions, which lists the sixteen possible binary truth functions of which eight are commutative: Vpq = Vqp; Apq (OR) = Aqp; Dpq (NAND) = Dqp; Epq (IFF) = Eqp; Jpq = Jqp; Kpq (AND) = Kqp; Xpq (NOR) = Xqp; Opq = Oqp.
Concatenation, the act of joining character strings together, is a noncommutative operation. For example,
EA + T = EAT ≠ TEA = T + EA
Washing and drying clothes resembles a noncommutative operation; washing and then drying produces a markedly different result to drying and then washing.
Rotating a book 90° around a vertical axis then 90° around a horizontal axis produces a different orientation than when the rotations are performed in the opposite order.
Thought processes are noncommutative: A person asked a question (A) and then a question (B) may give different answers to each question than a person asked first (B) and then (A), because asking a question may change the person's state of mind.
The act of dressing is either commutative or non-commutative, depending on the items. Putting on underwear and normal clothing is noncommutative. Putting on left and right socks is commutative.
Shuffling a deck of cards is non-commutative. Given two ways, A and B, of shuffling a deck of cards, doing A first and then B is in general not the same as doing B first and then A.
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 (orcross 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. The word is a combination of the French word commuter meaning "to substitute or switch" and the suffix -ative meaning "tending to" so the word literally means "tending to substitute or switch". The term then appeared in English in 1838[2]inDuncan Farquharson 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
Ingroup 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.
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 graphinthree-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 .
Inquantum 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 principleofHeisenberg, 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.