Módulo

Módulo is a term used in several languages, notably Portuguese and Spanish, with multiple meanings in mathematics, computing, and engineering. It can denote a modulus, which is a size-related value or a remainder, or a self-contained unit such as a software or hardware module.

In mathematics, a modulus may refer to several concepts. The modulus of a complex number z = a + bi is |z| = sqrt(a^2 + b^2), representing its distance from the origin. The modulo operation, for integers a and n > 0, yields the remainder r = a mod n with 0 ≤ r < n. Congruence, written a ≡ b (mod n), holds when n divides a − b. Modular arithmetic is the arithmetic of residues modulo n and is central to number theory and cryptography, with results such as Fermat’s and Euler’s theorems and tools like the Chinese remainder theorem.

In algebra, a module is a generalization of a vector space in which scalars come from a ring rather than a field. This abstraction underpins many areas of mathematics and includes structures such as abelian groups viewed as modules over the ring of integers Z.

In computing and engineering, a module is a self-contained component with a defined interface or function. Software modules are units of code that can be reused or replaced, while hardware modules are detachable units in modular systems, such as plug-in cards or modular devices, designed to be combined to form larger systems.