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.
- hyper^{4} is symbolized by either a^^b or a^{(4)}b, and is defined as a^{(4)}b = a^(a^(...^a)) = a^(a^(b-1))).
- hyper_{4} is symbolized by a _{(4)} and is defined as a_{(4)}b = ((a^a)^...)^a.
- Only the former, hyper^{4}, definition is technically a different operator, since the hyper_{4} operation can be reduced to exponentiated exponentiation (iterated exponentiation).
- hyper_{5} = a^^^b = a_{(5)}b = a^{(4)}a^{(4)}...a^{(4)}
- hyper_{6} = 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 d^{n}/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:
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