nonzero

Nonzero is a mathematical term used to describe an element that is not equal to the designated zero in a given structure. The meaning of zero depends on context: in an additive group, ring, or module, the zero element acts as the additive identity; in a multiplicative setting such as a field, the element 1 acts as the multiplicative identity and the term nonzero typically excludes 0.

In vector spaces, a vector is nonzero if it is not the zero vector. In linear algebra, the set of nonzero vectors is the complement of {0}. A nonzero polynomial is a polynomial with at least one nonzero coefficient, i.e., not the zero polynomial. A nonzero function is a function that is not identically zero; there exists some input for which the function value is not zero. A nonzero matrix is a matrix different from the zero matrix, and its determinant may indicate invertibility.

In a field, the nonzero elements form a multiplicative group, often denoted by F* or K*. These elements each have a multiplicative inverse. The concept of nonzero is essential in solving equations, determining invertibility, and analyzing zeros of functions or polynomials. The term is relative to the chosen zero element, so careful specification of the structure is required.