# commutativity

Commutativity is a widely used mathematical term that refers to the ability to change the order of something without changing the end result. It is a fundamental property in most branches of mathematics and many proofs depend on it. 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.

## 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:
xy = yx for every x,yS
• An operation that does not satisfy the above property is called noncommutative.
2. One says that x commutes with y under ∗ if:
xy = yx

3. A binary function f:A×AB is said to be commutative if:
f(x,y) = f(y,x) for every x, yA.

## History

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 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.[11]

## Related Properties

### 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. 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.

## Examples

### Commutative operations in everyday life

• Putting your shoes on resembles a commutative operation since it doesn't matter if you put the left or right shoe on first, the end result (having both shoes on), is the same.
• When making change we take advantage of the commutativity of addition. It doesn't matter what order we put the change in, it always adds to the same total.

### Commutative operations in math

Two well-known examples of commutative binary operations are:[12]
:
For example 4 + 5 = 5 + 4, since both expressions equal 9.
:
For example, 3 Ã— 5 = 5 Ã— 3, since both expressions equal 15.

### Noncommutative operations in everyday life

• Washing and drying your clothes resembles a noncommutative operation, if you dry first and then wash, you get a significantly different result than if you wash first and then dry.
• The Rubik's Cube is noncommutative. For example, twisting the front face clockwise, the top face clockwise and the front face counterclockwise (FUF') does not yield the same result as twisting the front face clockwise, then counterclockwise and finally twisting the top clockwise (FF'U). The twists don't commute. This is studied in group theory.

### Noncommutative operations in math

Some noncommutative binary operations are:[13]

## 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. ^ CabillÃ³n and Miller, Commutative and Distributive
10. ^ O'Conner and Robertson, Servois
11. ^ CabillÃ³n 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

• Axler, Sheldon (1997). Linear Algebra Done Right, 2e. Springer. ISBN 0-387-98258-2.
''Abstract algebra theory. Covers commutativity in that context. Uses property throughout book.
• Goodman, Frederick (2003). Algebra: Abstract and Concrete, Stressing Symmetry, 2e. Prentice Hall. ISBN 0-13-067342-0.
''Abstract algebra theory. Uses commutativity property throughout book.
• Gallian, Joseph (2006). Contemporary Abstract Algebra, 6e. ISBN 0-618-51471-6.
Linear algebra theory. Explains commutativity in chapter 1, uses it throughout.

### Articles

Article describing the mathematical ability of ancient civilizations.
• Robins, R. Gay, and Charles C. D. Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text. London: British Museum Publications Limited. ISBN 0-7141-0944-4
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

Commute or Commutation may refer to:
• Commuting, the process of travelling between a place of residence and a place of work
• Commutativity, a property of a mathematical operation
• Commutation of sentence, a reduction in severity of punishment

In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator.
Analysis has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function.
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations.
Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. Addition also provides a model for related processes such as joining two collections of objects into one collection.
Multiplication is the mathematical operation of adding together multiple copies of the same number. For example, four multiplied by three is twelve, since three sets of four make twelve:

In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator.
In mathematics, a binary function, or function of two variables, is a function which takes two inputs.

Precisely stated, a function is binary if there exists sets such that
Where is the Cartesian product of and

For example, if Z
Gumhūriyyat Miṣr al-ʿArabiyyah
Arab Republic of Egypt

Flag Coat of arms
Anthem
Euclid

Born fl. 300 BC

Residence Alexandria, Egypt
Nationality Greek
Field Mathematics
Known for Euclid's Elements Euclid (Greek:
Euclid's Elements (Greek: Στοιχεῖα) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC.
associativity is a property that a binary operation can have. It means that, within an expression containing two or more of the same associative operators in a row, the order of operations does not matter as long as the sequence of the operands is not changed.
Symmetry in mathematics occurs not only in geometry, but also in other branches of mathematics. It is actually the same as invariance: the property that something does not change under a set of transformations.
Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. Addition also provides a model for related processes such as joining two collections of objects into one collection.
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
expression must be well-formed. That is, the operators must have the correct number of inputs, in the correct places. The expression 2 + 3 is well formed; the expression * 2 + is not, at least, not in the usual notation of arithmetic.
Multiplication is the mathematical operation of adding together multiple copies of the same number. For example, four multiplied by three is twelve, since three sets of four make twelve:

In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
In mathematics, a complex number is a number of the form

where a and b are real numbers, and i is the imaginary unit, with the property i Â² = −1.
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.
In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else.

## Basic definition

If A and B are sets, then the union of A and B
SET may stand for:
• Sanlih Entertainment Television, a television channel in Taiwan
• Secure electronic transaction, a protocol used for credit card processing,

Rubik's Cube (commonly misspelled rubix, rubick's or rubicscube) is a mechanical puzzle invented in 1974[1] by the Hungarian sculptor and professor of architecture Ernő Rubik.
Group theory is the mathematical study of symmetry, as embodied in the structures known as groups. These are sets with a closed binary operation satisfying the following three properties:
1. The operation must be associative.
2. There must be an identity element.

Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. Subtraction is denoted by a minus sign in infix notation.

The traditional names for the parts of the formula
cb = a
are