naturals0

Naturals0 is the designation used for the set of natural numbers that includes zero. In mathematical notation it is commonly written as N0, ℕ0, or ℕ ∪ {0}. Depending on the source, the symbol ℕ may denote either {0,1,2,...} or {1,2,3,...}, so N0 explicitly specifies the version that includes zero.

Definition and scope. N0 consists of the nonnegative integers: {0,1,2,3,...}. It is a subset of the integers Z. In some contexts, ℕ denotes {1,2,3,...}, in which case N0 is used to emphasize the inclusion of zero.

Algebraic properties. The set N0 is closed under addition and multiplication. It contains the additive identity 0 and the multiplicative identity 1. Subtraction is not closed within N0, since results can be negative; division is not closed either. The factorial function is defined for all n ≥ 0, which makes N0 convenient in combinatorics and number theory.

Usage. In mathematics, N0 is used to describe counting with zero included, indexing in certain formulas, and many proofs where the nonnegative integers provide a convenient domain. In computer science, N0 aligns with unsigned integers and with zero-based indexing in many programming languages, where the first element of an array has index 0.

Terminology and variants. Some authors distinguish between natural numbers with zero (N0) and without zero (N = {1,2,...}); others simply use ℕ with an agreed convention. The term naturals0 is informal and primarily used in programming or documentation to denote nonnegative integers.