Operators versus functions edit
The word operator can in principle be applied to any function. However, in practice it is most often applied to functions that operate on mathematical entities of higher complexity than real numbers, such as vectors, random variables, or mathematical expressions. The differential and integral operators, for example, have domains and codomains whose elements are mathematical expressions of indefinite complexity.
In general, if either the domain or codomain (or both) of a function contains elements significantly more complex than real numbers, that function is referred to as an operator. Conversely, if neither the domain nor the codomain of a function contain elements more complicated than real numbers, that function is likely to be referred to simply as a function. Trigonometric functions such as cosine are examples of the latter case.
Additionally, when functions are used so often that they have evolved faster or easier notations than the generic F(x,y,z,...) form, the resulting special forms are also called operators. Examples include infix operators such addition "+" and division "/", and postfix operators such as factorial "!". This usage is unrelated to the complexity of the entities involved.
Influences from other disciplines edit
Concepts from other disciplines, including in physics and to a lesser degree computer science, have also influenced the ways in which operators are perceived and used.
Physics edit
The mutual influence between physics and mathematics regarding the concept of operators has been long-term, beginning in the early 1900s, and profound in both directions. Quantum mechanics in particular was forced to move to mathematical approaches that replace classical measurements using simple numeric quantities with operators that transform between far less intuitive entities that include vectors in both ordinary space and abstract generalizations known as Hilbert spaces, spinors, and various forms of matrices. The great physicist P.A.M. Dirac captured the mutual importance of this relationship between quantum physics and mathematics by noting during during lecture in the U.S.S.R in 1955 that "Physical laws should have mathematical beauty and simplicity."[1]
Computer science edit
Computer science has contributed the concept of overloading to operators. An overloaded operator is one for which the same function notation is interpreted differently depending on the type signature) of its arguments. Mathematically, an overloaded operator is itself an operator or "meta-operator" that maps from function labels and argument types into definite functions with a well-specified domains and codomains. Since overloading is meaningful only if an operator-preserving homomorphism exists for the different types of arguments, it is most commonly used with arguments that meet the mathematical definition of fields—that is, they all support addition, subtraction, multiplication, and division (except by zero).
For example, "+" can be defined through overloading to mean addition of integers, real numbers, matrices, and any meaningful combination thereof. This same example also shows the dangers of overloading, since every mapping of "+" and its possible argument types must be defined precisely.