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Example showing the commutativity of addition (3 + 2 = 2 + 3)
Example showing the commutativity of addition (3 + 2 = 2 + 3)
For other uses, see Commute (disambiguation).

In mathematics, commutativity is the ability to change the order of something without changing the end result. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and It is a fundamental property in most branches of mathematics and many proofs depend on it. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true The commutativity of simple operations was for many years implicitly assumed and the property was not given a name or attributed until the 19th century when mathematicians began to formalize the theory of mathematics.

Contents

Common uses

The commutative property (or commutative law) is a property associated with binary operations and functions. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function 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. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. 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. Analysis has its beginnings in the rigorous formulation of Calculus. Linear algebra is the branch of Mathematics concerned with Addition is the mathematical process of putting things together [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:

x ∗ y = y ∗ x for every x,y ∈ S

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:

f(x,y) = f(y,x) for every x, y ∈ A. In Mathematics, a binary function, or function of two variables, is a function which takes two inputs

History and etymology

The first known use of the term was in a French Journal published in 1814
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. This article is about the country of Egypt For a topic outline on this subject see List of basic Egypt topics. [6][7] Euclid is known to have assumed the commutative property of multiplication in his book Elements. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek [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 use of the actual term commutative was in a memoir by Francois 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. The Philosophical Transactions of the Royal Society, or Phil Trans [11]

Related properties

Graph showing the symmetry of the addition function
Graph showing the symmetry of the addition function

Associativity

Main article: associativity

The associative property is closely related to the commutative property. In Mathematics, associativity is a property that a Binary operation can have The associative property states that the order in which operations are performed does not affect the final result. In contrast, the commutative property states that the order of the terms does not affect the final result.

Symmetry

Main article: symmetry in mathematics

Symmetry can be directly linked to commutativity. Symmetry in Mathematics occurs not only in Geometry, but also in other branches of mathematics 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.

Examples

Commutative operations in everyday life

Commutative operations in math

Example showing the commutativity of multiplication (3 × 5 = 5 × 3)
Example showing the commutativity of multiplication (3 × 5 = 5 × 3)

Two well-known examples of commutative binary operations are:[12]

y + z = z + y \quad \forall y,z\in \mathbb{R}
For example 4 + 5 = 5 + 4, since both expressions equal 9. Addition is the mathematical process of putting things together In Mathematics, the real numbers may be described informally in several different ways In mathematics the word expression is a term for any well-formed combination of mathematical symbols
y z = z y \quad \forall y,z\in \mathbb{R}
For example, 3 × 5 = 5 × 3, since both expressions equal 15. In Mathematics, the real numbers may be described informally in several different ways

Noncommutative operations in everyday life

Concatenation, the act of joining character strings together, is a noncommutative operation.
Concatenation, the act of joining character strings together, is a noncommutative operation. For concatenation of general lists see Append. In Computer programming, string concatenation is the operation of joining two character

Noncommutative operations in math

Some noncommutative binary operations are:[13]

\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

Notes

  1. ^ Axler, p. 2
  2. ^ Gallian, p. 34
  3. ^ p. 26,87
  4. ^ 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. ^ Cabillón and Miller, Commutative and Distributive
  10. ^ O'Conner and Robertson, Servois
  11. ^ Cabillón and Miller, Commutative and Distributive
  12. ^ Krowne, p. 1
  13. ^ Yark, p. 1
  14. ^ Gallian, p. 34
  15. ^ Gallian p. 236
  16. ^ Gallian p. 250
  17. ^ Gallian p. 65

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. The Rhind Mathematical Papyrus (RMP (also designated as papyrus British Museum 10057 and pBM 10058 is named after Alexander Henry Rhind, a Scottish

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

See also

In mathematics anticommutativity refers to the property of an operation being anticommutative, i In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In Algebra, the commutant of a Subset S of a Semigroup (such as an algebra or a group) A is the subset In Neurophysiology, commutative has much the same meaning as in algebra. In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law Particle statistics refers to the particular description of particles in Statistical mechanics. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion.

Dictionary

commutativity

-noun

  1. (mathematics) (physics) The state of being commutative.
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