Mathematical Operation
A Mathematical Operation is an operation within a mathematical system.
- Context:
- It can range from being an Abstract Mathematical Operation to being a Mathematical Software Operation.
- It can be represented by a Mathematical Symbol (e.g. ×).
- It can be a member of a Mathematical Operation Set.
- It can be a Logic Operation.
- It can be used in a Mathematical Expression (to specify a computation).
- Example(s):
- an Arithmetic Operation.
- an Algebraic Operation.
- a Vector Operation, Matrix Operation.
- (5+5)-4 ⇒ 5+(5-4).
- 1/2+1/3 ⇒ 3/6 + 2/6 ⇒ 5/6.
- a Summation Operation.
- a Multiplication Operation.
- …
- Counter-Example(s):
- See: Mathematical Theorem.
References
2014
- (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Operation_(mathematics) Retrieved:2014-4-26.
- The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation.
In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values, called “operands”. There are two common types of operations: unary and binary. Unary operations involve only one value, such as negation and trigonometric functions. Binary operations, on the other hand, take two values, and include addition, subtraction, multiplication, division, and exponentiation.
Operations can involve mathematical objects other than numbers. The logical values true and false can be combined using logic operations, such as and, or, and not. Vectors can be added and subtracted. Rotations can be combined using the function composition operation, performing the first rotation and then the second. Operations on sets include the binary operations union and intersection and the unary operation of complementation. Operations on functions include composition and convolution.
Operations may not be defined for every possible value. For example, in the real numbers one cannot divide by zero or take square roots of negative numbers. The values for which an operation is defined form a set called its domain. The set which contains the values produced is called the codomain, but the set of actual values attained by the operation is its range. For example, in the real numbers, the squaring operation only produces nonnegative numbers; the codomain is the set of real numbers but the range is the nonnegative numbers.
Operations can involve dissimilar objects. A vector can be multiplied by a scalar to form another vector. And the inner product operation on two vectors produces a scalar. An operation may or may not have certain properties, for example it may be associative, commutative, anticommutative, idempotent, and so on.
The values combined are called operands, arguments, or inputs, and the value produced is called the value, result, or output. Operations can have fewer or more than two inputs.
An operation is like an operator, but the point of view is different. For instance, one often speaks of "the operation of addition" or "addition operation" when focusing on the operands and result, but one says "addition operator" (rarely "operator of addition") when focusing on the process, or from the more abstract viewpoint, the function +: S×S → S.
- The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation.
2009
- (WordNet, 2009) ⇒ http://wordnetweb.princeton.edu/perl/webwn?s=operation
- S: (n) mathematical process, mathematical operation, operation ((mathematics) calculation by mathematical methods) "the problems at the end of the chapter demonstrated the mathematical processes involved in the derivation"; "they were learning the basic operations of arithmetic"
- http://en.wiktionary.org/wiki/operation
- 4. (mathematics) a procedure for generating a value from one or more other values (the operands; the value for any particular operands is unique)
- http://planetmath.org/encyclopedia/Operation.html
- Examples
- 1. Arithmetic operations: addition, subtraction, multiplication, division. Their generalization leads to the so-called binary operations, which is a basic concept for such algebraic structures as group, ring, field.
- 2. Operations on vectors in the plane ($ \mathbb{R}^2$).
- Multiplication by a scalar. Generalization leads to vector spaces.
- Scalar product. Generalization leads to Hilbert spaces.
- 3. Operations on vectors in the space ($ \mathbb{R}^3$).
- Cross product. Can be generalized for the vector space of arbitrary finite dimension, see vector product in general vector spaces.
- Triple product.
- 4. Some operations on functions.
- Composition.
- Function inverse.
- http://planetmath.org/encyclopedia/Expression.html
- operator: a mathematical symbol such as '+', '-', '*', and '/' used as part of a mathematical expression to specify a computation operation such as addition, subtraction, multiplication and division respectively. See also boolean operator, and string expression