Operator

In Mathematics, an operator is a symbol indicating an operation to be performed on one or more operands.

The following elementary binary/dyadic arithmetic operators are quite standard:

• addition symbolized by '+' as in 1+1=2.
• subtraction symbolized by '-' as in 2-1=1.
• multiplication symbolized by '×' as in 2 × 3 = 6, or by simple juxtaposition as in xy for the product of x and y.
• division symbolized by '/', '÷' or a horizontal line separating numerator from denominator as in 3/2=1.5 .
• exponentiation n^m by elevation of the exponent m above the base line. If the exponent m is a positive integer, then the exponent describes the number of factors (repeated multiplication)

Past these basic operations lie the hyper-n operators

• hyper4, also known as tetration, superpower, superdegree, or powerlog.
  • hyper⁴ is symbolized by either a^^b or a⁽⁴⁾b, and is defined as a⁽⁴⁾b = a^(a^(...^a)) = a^(a^(b-1))).
  • hyper₄ is symbolized by a ₍₄₎ and is defined as a₍₄₎b = ((a^a)^...)^a.
  • Only the former, hyper⁴, definition is technically a different operator, since the hyper₄ operation can be reduced to exponentiated exponentiation (iterated exponentiation).
• hyper₅ = a^^^b = a₍₅₎b = a⁽⁴⁾a⁽⁴⁾...a⁽⁴⁾
• hyper₆ = a^^^^b = a₍₆₎b = a⁽⁵⁾a⁽⁵⁾...a⁽⁵⁾
• ad infinitum[?].

These can be written equivalently using Knuth's up-arrow notation.

The hyper-n concept also extends into trinary/triadic operators.

• addition = hy(a,1,b)
• multiplication = hy(a,2,b)
• exponentiation = hy(a,3,b)
• hyper4 = hy(a,4,b)

Different branches of mathematics may extend the definitions of operators to represent analogous operations.

• The concept of an addition operator '+' has been extended to cover addition of sets, vectors and matrices.
• Multiplication of a vector by a particular matrix is a unary operator or transformation; it is common, and only a slight abuse of language, to say the matrix is the operator.
• Operators for mathematical functions: '+' defines the sum f+g of two functions f and g by (f+g)(x)=f(x)+g(x); similar f-g, f*g, f/g, f^g. Additionally, other operators are possible, e.g., function composition: f o g = f(g) defined by (f(g))(x)=f(g(x)); convolution which is defined by an integral.
• Differential operators such as d/dx (notationally equivalent forms are the n-th derivative dⁿ/dx, Heaviside's Big D operator), the Laplacian, the divergence, the gradient, the curl, Sturn-Liouville operators, etc.
• Integral operators of the form

<math>(Tf)(y)=\int_A f(x)k(x,y)\,dx</math>

including such things as the Fourier and Laplace transforms.

• Operators of probability theory such as expectation, variance, covariance, etc.
• Operators are a key part of the theory of quantum mechanics

Linear operators are those which satisfy the following conditions; take the general operator Q, and the constant a:

<math>Q(f(x)+g(x)) = (Qf)(x)+(Qg)(x)</math>

<math>(Qf)(ax) = a(Qf)(x)</math>

Such examples of linear operators are the differential and Laplace transforms.

This is a stub article and needs much work. May I suggest to those who considered moving it to "Mathematical operator" that "Operator (mathematics)" would be a better name. The reason for that is that mathematicians say "operator" without often putting the word "mathematical" in front of it.

See also:

• function, unary operation, binary operation, trinary operation[?].