Multiplicative

Multiplicative is an adjective derived from multiplication. In mathematics, it describes objects, properties, or processes that involve multiplication or that preserve the product operation. The term is used in several related senses, often distinguished by how strictly they interact with multiplication.

In number theory, a function is called multiplicative if it respects multiplication for coprime arguments: f(1) = 1 and f(ab) = f(a)f(b) whenever a and b are coprime. If the same rule holds for all a and b, the function is completely multiplicative. Examples include the constant function f(n) = 1, the power functions f(n) = n^k, Euler’s totient phi(n), the Möbius function mu(n), and the divisor function d(n). Multiplicative functions have important structural properties and often arise in formulas that factor over primes. The Dirichlet convolution of multiplicative functions is also multiplicative.

In algebra, multiplicativity can describe maps that preserve multiplication. A ring homomorphism preserves both addition and multiplication, reflecting multiplicativity of the product operation. A multiplicative monoid is a set equipped with an associative multiplication and an identity element, focusing on the product structure rather than additive aspects.

Outside pure theory, multiplicative effects are used in modeling to indicate that changes scale by a product rather than by addition. For example, a multiplicative effect multiplies a baseline quantity, producing growth or decline proportional to the baseline.

Notes: the phrase is context-dependent. In logic, “multiplicative” can refer to a specific fragment with resource-sensitive connectives; such uses are specialized beyond standard arithmetic and algebra.