modulus

Modulus is a term used in mathematics to denote a measure of size or magnitude. Its precise meaning varies with context, but it commonly refers to absolute value, the magnitude of a complex number, or the divisor used in modular arithmetic.

For a real number x, the modulus is its absolute value, denoted |x|. This value is non‑negative and equals x when x ≥ 0, while it equals −x when x < 0. It satisfies properties such as |xy| = |x| |y| and |x+y| ≤ |x| + |y|.

For a complex number z = a + bi, the modulus is |z| = sqrt(a^2 + b^2). Geometrically, it is the distance from the origin in the complex plane. The complex number can be expressed in polar form z = |z| e^{iθ}, where θ is the argument of z.

In modular arithmetic, the modulus is the positive integer n used to define congruence. Two integers a and b are congruent modulo n if n divides their difference. The remainder upon division by n is often written as a mod n, satisfying 0 ≤ a mod n < n. Modulus plays a central role in number theory and is widely used in cryptography, such as RSA, which relies on arithmetic modulo large primes or products of primes.

Etymology traces modulus to Latin modulus, meaning a small measure. In programming contexts, modulus also appears as the modulus or remainder operation, with implementation details that may vary for negative inputs across languages.