In mathematics, an operator generally is a symbolism to show a certain mapping, usually from one or more given functions to another (between function spaces), however, operators can refer to mappings between vector spaces in general as well.
Operators in mathematics
Operators generally transform functions into other functions. We also say an operator maps a function to another. In some literature, they are designated by showing a small uphat over the operator name. In certain circumstances, they are written unlike functions, when an operator has a single argument or operand, for example, if the operator name is called Q and operates on a function f, we write Qf and not usually Q(f), however this latter notation may be used for clarity if there is a product for instance, eg. Q(fg). Throughout this article we will use Q to denote a general operator, and x_{i} to denote the i-th argument.Notations for operations on functions may also be notated as the following. If f(x) is a function of x and Q is the general operator we can write Q acting on f as:
- (Qf)(x)
Functions can be considered operators, but are generally thought of differently conceptually. "Numbers" can be considered functions too, if f(x)=x^{0}, this represents the number 1. Similarly after multiplication by a constant, any number can be defined. When an operator takes some numbers as arguments, we can consistenly regard the operator as still transforming functions, since we have seen that numbers can be considered as functions.
Notations and ideas
Describing operators
Operators are described usually by the number of operands:- monodic, or unary: one argument
- dyadic, or binary: two arguments
- triadic, or ternary: three arguments
Notating operators
There are three major ways of writing operators and their arguments. These are- prefix: where the operator name comes first and the arguments follow, for example:
- postfix: where the operator name comes last and the arguments precede, for example:
- infix: where the operator name comes between the arguments. This is not commonly used for operators taking greater than 2 arguments, ie binary operators. Trivially for an operator taking 1 argument, writing infix is equivalent to writing prefix. Infix style is written, for example:
Linear operators
A key concept is the concept of the linear operator. Linear operators are those which satisfy the following conditions; take the general operator Q, the function acted on under the operator Q, written as f(x), and the constant a:Linear operators with respect to mappings between vector spaces are known more commonly as linear transformations or linear mappings.
Such an example of a linear transformation between vectors in R^{2} is reflection, given a vector x=(x_{1}, x_{2})
- Q(x_{1}, x_{2})=(-x_{1}, x_{2})
Additive operators
An additive operator, in abstract algebra, may satisfy the commutative and associative laws. If there is also a predefined multiplicative operator then the operator must satisfy the distributive law.Multiplicative operators
A multiplcative operator, in abstract algebra, may satisfy the associative law. If there is also a predefined multiplicative operator the operator must satisfy the distributive law.Standard operators
Arithmetic operators are binary operators that perform simple transformations that many would find familiar. It is not obvious, but addition, subtraction, etc. are in fact operators. Many of these standard arithmetic operators use symbols to denote what operations are being performed.Addition
Addition is written using the symbol +. It transforms two numbers x_{1} and x_{2} into their sum. For example:- 3 + 5 = 8
- plus x_{1} x_{1}.
Subtraction
Subtraction is written using the symbol -. It transforms two numbers x_{1} and x_{2} into their difference. For example:- 11 - 4 = 7
- sub x_{1} x_{1}.
- x_{1} - x_{2} ≡ x_{1} + (-x_{2})
Negation
Negation is written also using the symbol -, however, it is only a unary operator. Given a number α, we denote the transformation of α to its additive inverse by -α. The additive inverse of a number k is an element k', such that k+k'=0.Multiplication
Multiplication is written using the symbol ×. In certain circumstances, the operator symbol is omitted usually when the arguments to × are variable quantities, eg xy. Less commonly when representing the product of two numbers, they are placed in brackets and placed adjacently, eg. (2)(3)=6. Less commoner still, a small dot is used infix instead of ×, eg 2·3=6Multiplication transforms two numbers x_{1} and x_{2} into their product. For example:
- 6 × 2 = 12
- mul x_{1} x_{1}.
- x_{1}x_{2} ≡ x_{1} + x_{1} + ...( x_{}2 times)...+x_{1}
Division
Division is written using the symbol /. Like multiplication, there are several ways to denote this, other than using /. If there is not much room on a page, or when typeset on a single line, the two arguments are written infix, eg 3 / 4, or x_{1}/x_{2}. If there is room on a page, the two arguments are usually written atop each other and a line seperating them, eg:- 8 / 2 = 4
- div x_{1} x_{1}.
Exponentiation
Exponentiation is most generally not written using a symbol, but with the second argument written as a superscript, for example . In certain circumstances, as in representing this operation in programming, the symbol ^ is used.Exponentiation transforms two numbers x_{1} and x_{2} into their repeated product. For example:
- 6^{2} = 6 × 6 = 36
- pow x_{1} x_{1}.
- x_{1}x_{2} ≡ x_{1}x_{1} ...( x_{}2 times)...x_{1}.
Generalizations
With addition as a basic operator, we can see that the extension of multiplication is an iterated addition. Exponentiation is an iterated multiplication.We have a notation we can use to show an extension of this generality.
hyper_{4} is the operator that is defined as repeated exponentiation. If we define Q to be a binary operator, Q x_{1} x_{2} =
This operation has several names, viz., tetration, superpower, superdegree, or powerlog. The two most common notations for this is Knuth's up-arrow notation as x_{1} ↑↑ x_{2}, and hyper^{4}. Less commonly seen, though somewhat more convenient notations are x_{1}^{(4)}x_{2}.
Only the hyper^{4}, definition is technically a different operator, since this operation can be reduced to exponentiated exponentiation (iterated exponentiation). If we again define Q x_{1} x_{2} = : as before, then we define x_{1} ↑↑↑ x_{2} or hyper_{5}(x_{1},x_{2}) as being:
Further generalizations can be taken similarly ad infinitum.
We can generalize back addition, multiplication, and exponentiation in terms using the notations we have just described, ie.,
- hyper_{1} x_{1} x_{2} = x_{1} + x_{2}
- hyper_{2} x_{1} x_{2} = x_{1}x_{2}
- hyper_{3} x_{1} x_{2} =
Similar behaviors
Some operators aforementioned can also have other behaviours than what was previously described. In programming terms, this is known as overloading, however in mathematics the meaning of an operation is understood from the context by generally the subject matter or what the arguments are. Some examples follow.Addition operator
The concept of the addition operator + has been extended to cover addition of sets, vectorss and matrices.Matrix multiplication
Multiplication of a vector by a particular matrix is a unary operator or transformation. We can regard the multiplication of the matrix to be an operator (see below).Elementary function operators
We have seen that an operator transforms one function to another. So, we can define + to be the sum of the two functions, x_{1} and x_{2}. resulting in another function. For example, if we define Q this way;- Q (x^{2}+3x) (5x^{2}+9) = 6x^{2}+3x+9
Function composition
Additionally, we have some other operators which we can define on functions. One such fundamental operator is that of function composition. Given two functions x_{1}=f(t) and x_{2}=g(t), define the operator Q:- Q x_{1} x_{2} = f(g(t))
- f(g(t))=(fog)(x)=x_{1} o x_{2}
Probability theory
Operators are also involved in probability theory. Such operators as expectation, variance, covariance, factorials, et al.Factorials are essential to the combination and permutation functions of probability and combinatorics, and are also the most commonly known postfix operator, being denoted by a ! placed after the number it expands. Its expansion follows the pattern,
- x! = 1 * 2 * ... * (x-1) * x
Calculus and operators
Calculus is, essentially, the study of one particular operator, and its behavior embodies and exemplifies the idea of the operator in great clarity. This key operator we study in Calculus is the differential operator.The differential operator
The differential operator is the symbolism used in Calculus to denote the action of taking a derivative. Common notations are such d/dx, y'(x) to denote the derivative of y(x). However here we will use the notation that is closest to the operator notation we have been using, that is, using D f to represent the action of taking the derivative of f.Notations
If f is a function of n variables t_{1},t_{1},...,t_{n}, we writeThe act of integration is also equivalent somewhat to taking the derivative backwards. So, in a sense it is differentiating -1 times, so we have integration in terms of the differential operator:
Integral operators
Given that integration is an operator as well, we have some important operators we can write in terms of integration.Convolution
The convolution of two functions is a mapping from two functions to one other, defined by an integral as follows:If x_{1}=f(t) and x_{2}=g(t), define the operator Q such that;
Fourier transform
The Fourier transform is another integral operator, and is used in many areas, not only in mathematics, but in physics and in signal processing, to name a few.It is based on the theorem that any continuous periodic function can be represented as the sum of a series of sine waves:
f(x) = ∑ A_{1} sin ω + A_{2} sin ω/2 + A_{3} sin ω/3 + ...
See Fourier transform for more information.
Laplacian transform
The Laplacian transform is another integral operator and is involved in simplifying the process of solving differential equations.Given f=f(s), it is defined by:
Operators in physics
In physics, an operator often takes on a more specialized meaning that in mathematics. It often means a linear transformation from a Hilbert space to another or an element of a C* algebra. See Operator (physics).Operators are also a key part of the theory of quantum mechanics.
Operators in programming
The arithmetic operators are the same as the mathematical ones while the bit (binary digit) operations deal with the binary number system. The logical operators determine boolean values. The string operators manipulate strings of text and there are operators which allocate segments of memory for use.
Operators are also terms for some functionality in programming languages. Consider the C programming language syntax for pointers, using the operators * and &. sizeof is sometimes considered an operator, and in C++, new and delete are also operators.
In object oriented languages, like C++, you can define your own uses for operators.
Operators in telecommunications
Operators in telecommunications, who are usually women, aid telephone users in various ways including long distance calling, directory assistance and telephone repair. As technology advances, human operators are becoming more often replaced by a computerized system, and the idiom is turning over to mean a secret agent.See also
- function, unary operation, binary operation, trinary operation.