The Legendre symbol is a completely multiplicative function of its top argument:
This is a time based model that describes the probability of a system operator's failure as a multiplicative function of 5 main factors.
Totally additive is also used in this sense by analogy with totally multiplicative functions.
A multiplicative function modeled after this phenomenon was created.
Many specific examples are given in the article on multiplicative functions.
Completely multiplicative functions also satisfy a pseudo-associative law.
Outside number theory, the term multiplicative function is usually used for completely multiplicative functions.
Some multiplicative functions are defined to make formulas easier to write:
With this operation, the set of all multiplicative functions turns into an abelian group; the identity element is .
Relations among the multiplicative functions discussed above include:
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