0×n

0×n denotes the product of zero and a quantity n. In standard arithmetic over the real numbers, integers, rational numbers, and most common mathematical structures, 0×n equals 0 for every n. When n is a variable, the expression represents the constant function that outputs zero for all allowed n.

A key property is that 0 is the absorbing element for multiplication: for any number x, x×0 = 0 and 0×x = 0 (when the operation is defined). This underpins many algebraic laws and is used to simplify equations by removing factors.

Examples include 0×7 = 0 and 0×(-3.14) = 0. In higher mathematics, the same idea extends to other contexts: in vector spaces, scalar multiplication by zero yields the zero vector; in matrix algebra, multiplying a matrix by the scalar 0 produces the zero matrix of the same dimensions.

In modular arithmetic, 0×n ≡ 0 (mod m) for any n and modulus m. The concept also arises in limits, where expressions like 0×∞ are indeterminate and require more careful analysis, rather than being treated as a definite value.

Historically, the recognition of zero as an absorbing element in multiplication is a standard part of the development of algebra and contributes to broader abstract algebra concepts, such as rings and fields. Overall, 0×n is a straightforward instance of the broader rule that multiplication by zero yields zero.