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 xi 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:
Functions can be considered operators, but are generally thought of differently conceptually. "Numbers" can be considered functions too, if f(x)=x0, 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.
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 R2 is reflection, given a vector x=(x1, x2)
Multiplication transforms two numbers x1 and x2 into their product. For example:
Notations and ideas
Describing operators
Operators are described usually by the number of operands:
and so on.Notating operators
There are three major ways of writing operators and their arguments. These are
Q(x1, x2,...,xn).
(x1, x2,...,xn) Q
x1 Q x2Linear 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:
Such examples of linear operators are the differential operator and Laplacian operator, which we will see later.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 x1 and x2 into their sum. For example:
It is written most commonly as x1+x2.
In prefix notation, it may be written as + x1 x2, or +(x1,x1), or even with + changed to a word, such as
Addition follows the field axioms.Subtraction
Subtraction is written using the symbol -. It transforms two numbers x1 and x2 into their difference. For example:
It is written most commonly as x1-x2.
In prefix notation, it may be written as - x1 x2, or -(x1,x1), or even with - changed to a word, such as
Subtraction is equivalent to addition. The identity is that:
where - as a unary operator represents negation (see next section)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=6
It is written most commonly as x1x2.
In prefix notation (using ×) it may be written as × x1 x2, or ×(x1,x1), or even with × changed to a word, such as
Multiplication is equivalent to repeated addition. The identity is that:
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 x1/x2. If there is room on a page, the two arguments are usually written atop each other and a line seperating them, eg:
Division transforms two numbers x1 and x2 into their quotient. For example:
It is written most commonly as x1/x2.
In prefix notation (using /) it may be written as / x1 x2, or /(x1,x1), or even with / changed to a word, such as
Division is equivalent to repeated subtraction. x2 is subtracted from x1 until there only is a positive remainder left. When, after application of this algorithm, there is zero remainder, we call the amount of subtractions we have performed the quotient. If not, we can write the result of this operation as either a fraction or as a decimal number (See those articles for further information).
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 x1 and x2 into their repeated product. For example:
We have a notation we can use to show an extension of this generality.
hyper4 is the operator that is defined as repeated exponentiation. If we define Q to be a binary operator, Q x1 x2 =
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 x1 ↑↑ x2, and hyper4. Less commonly seen, though somewhat more convenient notations are x1(4)x2.
Only the hyper4, definition is technically a different operator, since this operation can be reduced to exponentiated exponentiation (iterated exponentiation). If we again define Q x1 x2 = : as before, then we define x1 ↑↑↑ x2 or hyper5(x1,x2) 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.,
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).
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,
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, x1 and x2. resulting in another function. For example, if we define Q this way;
We can define multiplication, division, etc. in the same way. 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 x1=f(t) and x2=g(t), define the operator Q:
We write this operator infix using a small circle. So, with the same definitions as before,
Probability theory
Operators are also involved in probability theory.
Such operators as expectation, variance, covariance, factorials, et al.
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.
The 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:
If x1=f(t) and x2=g(t), define the operator Q such that;
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) = ∑ A1 sin ω + A2 sin ω/2 + A3 sin ω/3 + ...
See Fourier transform for more information.
Given f=f(s), it is defined by:
Operators are also a key part of the theory of quantum mechanics.
Programming languages, being that computers are mathematical devices, have a set of operators that perform various functions.
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.
Notations
If f is a function of n variables t1,t1,...,tn, we write
to represent the action of differentiating f with respect to ti.
If we differentiate f, k times, we write
How does the differential operator exemplify the idea of the operator?
Consider the function f=x2. Elementary calculus tells us that D f = 2x, futhermore if f=xα, D f = αxα-1. So we see clearly that the differential operator maps, or transforms, functions of the form xα to functions αxα-1.
It is clear that integration thus is equivalent to differentiation, so integration acts just like an operator as well -- mapping functions to functions.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:
which we write as .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.Laplacian transform
The Laplacian transform is another integral operator and is involved in simplifying the process of solving differential equations.
See Laplace transform for more information.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 in programming
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