nichtnegativer

Nichtnegativer is the German adjective used to describe quantities that are at least zero. In mathematics the term corresponds to the English nonnegative. A nonnegative value satisfies x ≥ 0, and the set of nonnegative real numbers is denoted by [0, ∞) on the real line. This concept extends to other mathematical objects, such as vectors, matrices, and functions, when every relevant component or value is required to be nonnegative.

In higher dimensions, the nonnegative orthant, often written as R^n_+, consists of all vectors (x1, x2, ..., xn) with xi ≥ 0 for every i. The idea of nonnegativity is central in areas such as probability (probabilities are nonnegative and sum to one), optimization (nonnegativity constraints), and linear algebra (nonnegative matrices or element-wise nonnegativity).

Key properties include that the sum and product of nonnegative numbers remain nonnegative. However, the difference of two nonnegative numbers need not be nonnegative. Nonnegativity is often extended to functions (nonnegative functions) and random variables (nonnegative almost surely). In calculus and analysis, nonnegativity provides important monotonic and convexity-related features.

Grammatical usage in German includes inflected forms of the adjective nichtnegativ. For example, ein nichtnegativer Wert means “a nonnegative value,” and der nichtnegative Wert means “the nonnegative value.” The term is common in technical and scientific writing to describe quantities that do not fall below zero.