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**. Binary operations are sometimes called **dyadic operations** in order to avoid confusion with the binary numeral system. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
The mathematical term arity sprang from words like unary, binary, ternary, etc. ...
In mathematics, a binary function, or function of two variables, is like a function, except that it has two inputs instead of one. ...
In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ...
The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ...
Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...
3 + 2 = 5 with apples, a popular choice in textbooks[1] Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. ...
5 - 2 = 3 Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ...
In mathematics, multiplication is an elementary arithmetic operation. ...
In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ...
More precisely, a binary operation on a set *S* is a binary function from *S* and *S* to *S*, in other words a function *f* from the Cartesian product *S* × *S* to *S*. Sometimes, especially in computer science, the term is used for any binary function. That *f* takes values in the same set *S* that provides its arguments is the property of closure. In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...
In mathematics, a binary function, or function of two variables, is like a function, except that it has two inputs instead of one. ...
In mathematics, the Cartesian product is a direct product of sets. ...
Computer scaence, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ...
In mathematics, a binary function, or function of two variables, is like a function, except that it has two inputs instead of one. ...
In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ...
Binary operations are the keystone of algebraic structures studied in abstract algebra: they form part of groups, monoids, semigroups, rings, and more. Most generally, a *magma* is a set together with any binary operation defined on it. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...
This picture illustrates how the hours in a clock form a group. ...
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...
In mathematics, a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation. ...
In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ...
In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ...
Many binary operations of interest in both algebra and formal logic are commutative or associative. Many also have identity elements and inverse elements. Typical examples of binary operations are the addition (+) and multiplication (*) of numbers and matrices as well as composition of functions on a single set. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
In mathematics, associativity is a property that a binary operation can have. ...
In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...
In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ...
3 + 2 = 5 with apples, a popular choice in textbooks[1] Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. ...
In mathematics, multiplication is an elementary arithmetic operation. ...
A number is an abstract idea used in counting and measuring. ...
In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...
In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...
Examples of operations that are not commutative are subtraction (-), division (/), exponentiation(^), and super-exponentiation(@). In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
5 - 2 = 3 Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ...
In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ...
Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n. ...
Tetration (also exponential map, hyperpower, power tower, super-exponentiation, and hyper4) is iterated exponentiation, the first hyper operator after exponentiation. ...
Binary operations are often written using infix notation such as *a* * *b*, *a* + *b*, or *a* · *b* rather than by functional notation of the form *f*(*a*,*b*). Sometimes they are even written just by juxtaposition: *ab*. Powers are usually also written without operator, but with the second argument as superscript. This article or section does not cite its references or sources. ...
Look up juxtaposition in Wiktionary, the free dictionary. ...
This article is about the term superscript as used in typography. ...
Binary operations sometimes use prefix or postfix notation, this dispenses with parentheses. Prefix notation is also called Polish notation; postfix notation, also called reverse Polish notation, is probably more often encountered. It has been suggested that this article or section be merged with Reverse Polish notation. ...
Postfix notation is a mathematical notation wherein every operator follows all of its operands. ...
## pair and tuple
A binary operation, *ab*, depends on the ordered pair (*a*,*b*) and so (*ab*)*c* (where the parentheses here mean first operate on the ordered pair (a,b) and then operate on the result of that using the ordered pair ((ab),c)) depends in general on the ordered pair ((*a*,*b*),*c*). Thus, for the general, non-associative case, binary operations can be represented with binary trees. In mathematics, an ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element (the first and second elements are also known as left and right projections). ...
In computer science, a binary tree is a tree data structure in which each node has at most two children. ...
However: - If the operation is associative, (
*ab*)*c*=*a*(*bc*), then the value depends only on the tuple (*a*,*b*,*c*). - If the operation is commutative,
*ab*=*ba*, then the value depends only on the multiset {{*a*,*b*},*c*}. - If the operation is both associative and commutative then the value depends only on the multiset {
*a*,*b*,*c*}. - If the operation is both associative and commutative and idempotent,
*aa*=*a*, then the value depends only on the set {*a*,*b*,*c*}. In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ...
In mathematics, a multiset (sometimes also called a bag) differs from a set in that each member has a multiplicity, which is a natural number indicating (loosely speaking) how many times it is a member, or perhaps how many memberships it has in the multiset. ...
In mathematics, a multiset (sometimes also called a bag) differs from a set in that each member has a multiplicity, which is a natural number indicating (loosely speaking) how many times it is a member, or perhaps how many memberships it has in the multiset. ...
In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ...
In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...
## External binary operations An **external binary operation** is a binary function from *K* and *S* to *S*. This differs from a binary operation in the strict sense in that *K* need not be *S*; its elements come from *outside*. An example of an external binary operation is scalar multiplication in linear algebra. Here *K* is a field and *S* is a vector space over that field. In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). ...
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. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
An external binary operation may alternatively be viewed as an action; *K* is acting on *S*. In mathematics, a symmetry group describes all symmetries of objects. ...
## See also |