Commutativity

Example showing the commutativity of addition (3 + 2 = 2 + 3)

In mathematics, commutativity is the property that changing the order of something does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. The commutativity of simple operations, such as multiplication and addition of numbers, was for many years implicitly assumed and the property was not named until the 19th century when mathematics started to become formalized.

Contents

Common uses

The commutative property (or commutative law) is a property associated with binary operations and functions. Similarly, if the commutative property holds for a pair of elements under a certain binary operation then it is said that the two elements commute under that operation.

In group and set theory, many algebraic structures are called commutative when certain operands satisfy the commutative property. In higher branches of math, 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.[1][2][3]

Mathematical definitions

The term "commutative" is used in several related senses.[4][5]

1. A binary operation ∗ on a set S is said to be commutative if:

\forall x,y \in S: x * y = y * x \,
- An operation that does not satisfy the above property is called noncommutative.

2. One says that x commutes with y under ∗ if:

x * y = y * x \,

3. A binary function f:A×A → B is said to be commutative if:

\forall x,y \in A: f (x, y) = f(y, x) \,

History and etymology

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

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.[6][7] Euclid is known to have assumed the commutative property of multiplication in his book Elements.[8] Formal uses of the commutative property arose in the late 18th and early 19th century 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. Simple versions of the commutative property are usually taught in beginning mathematics courses.

The first recorded use of the term commutative was in a memoir by François Servois in 1814,[9][10] 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 Philosophical Transactions of the Royal Society in 1844.[9]

Related properties

Graph showing the symmetry of the addition function

Associativity

Main article: associativity

The associative property is closely related to the commutative property. The associative property states that the order in which operations are performed does not affect the final result as long as the order of terms is not changed. In contrast, the commutative property states that the order of the terms does not affect the final result.

Symmetry

Symmetry can be directly linked to commutativity. When a commutative operator is written as a binary function then the resulting function is symmetric across the line y = x. As an example, if we let a function f represent addition (a commutative operation) so that f(x,y) = x + y then f is a symmetric function which can be seen in the image on the right.

For binary relations, a symmetric relation is analogous to a commutative operation, in that if a relation R is symmetric, then a R b \Leftrightarrow b R a.

Examples

Commutative operations in everyday life

Commutative operations in mathematics

Two well-known examples of commutative binary operations are:[4]

\forall (y,z) \in \mathbb{R}: y + z = z + y
For example 4 + 5 = 5 + 4, since both expressions equal 9.
\forall y,z\in \mathbb{R}: y z = z y
For example, 3 × 5 = 5 × 3, since both expressions equal 15.

Noncommutative operations in everyday life

EA + T = EAT \neq TEA = T + EA

Noncommutative operations in mathematics

Infinite addition is not (necessarily) commutative:

1-1+1-1+1-1+1-1+\dots<1

whereas

1+1-1+1+1-1+1+1-1+\dots=\infty

Some noncommutative binary operations are:[11]

\begin{bmatrix} 0 & 2 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix} \neq \begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix}

Mathematical structures and commutativity

Non-commuting operators in quantum mechanics

Main article: Uncertainty principle

In quantum mechanics as formulated by Schrödinger, physical variables are represented by linear operators such as x (meaning multiply by x), and d/dx. These two operators do not commute as may be seen by considering the effect of their products x (d/dx) and (d/dx) x on a one-dimensional wave function ψ(x):

x{d\over dx}\psi = x\psi' \neq {d\over dx}x\psi = \psi + x\psi'

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 that they cannot be simultaneously measured or known precisely. For example, the position and the linear momentum of a particle are represented respectively (in the x-direction) by the operators x and (h/2πi)d/dx (where h is Planck's constant). This is the same example except for the constant (h/2πi), 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

Notes

  1. ↑ Axler, p.2
  2. ↑ 2.0 2.1 Gallian, p.34
  3. ↑ p. 26,87
  4. ↑ 4.0 4.1 Krowne, p.1
  5. ↑ Weisstein, Commute, p.1
  6. ↑ Lumpkin, p.11
  7. ↑ Gay and Shute, p.?
  8. ↑ O'Conner and Robertson, Real Numbers
  9. ↑ 9.0 9.1 Cabillón and Miller, Commutative and Distributive
  10. ↑ O'Conner and Robertson, Servois
  11. ↑ Yark, p.1
  12. ↑ Gallian p.236
  13. ↑ Gallian p.250

References

Books

Abstract algebra theory. Covers commutativity in that context. Uses property throughout book.
Abstract algebra theory. Uses commutativity property throughout book.
Linear algebra theory. Explains commutativity in chapter 1, uses it throughout.

Articles

Article describing the mathematical ability of ancient civilizations.
Translation and interpretation of the Rhind Mathematical Papyrus.

Online resources

Definition of commutativity and examples of commutative operations
Explanation of the term commute
Examples proving some noncommutative operations
Article giving the history of the real numbers
Page covering the earliest uses of mathematical terms
Biography of Francois Servois, who first used the term