Encyclopedia > Operator

  Article Content

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 nm 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:



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
List of people by name: T

... Henry O.[?], (1859-1937), painter Tanner, Joseph[?], astronaut Tapia, Johnny, (born 1967), world boxing champion Tapscott, Horace[?], musician Tarantino, ...