(mathematics) Any function f that preserves addition; that is, f(x + y) = f(x) + f(y).
| Sci-Tech Dictionary: additive function |
(mathematics) Any function f that preserves addition; that is, f(x + y) = f(x) + f(y).
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| Wikipedia: Additive function |
Different definitions exist depending on the specific field of application. Traditionally, an additive function (or additive map) is a function that preserves the addition operation:
for any two elements x and y in the domain. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation.
In number theory, an additive function is an arithmetic function f(n) of the positive integer n such that whenever a and b are coprime, the function of the product is the sum of the functions:
The remainder of this article discusses number theoretic additive functions, using the second definition. For a specific case of the first definition see additive polynomial. Note also that any homomorphism f between Abelian groups is "additive" by the first definition.
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An additive function f(n) is said to be completely additive if f(ab) = f(a) + f(b) holds for all positive integers a and b, even when they are not coprime. Totally additive is also used in this sense by analogy with totally multiplicative functions.
Every completely additive function is additive, but not vice versa.
Arithmetic functions which are completely additive are:
From any additive function f(n) it is easy to create a related multiplicative function g(n) i.e. with the property that whenever a and b are coprime we have:
One such example is g(n) = 2f(n) − f(1).
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