Definition:Initial Segment of Natural Numbers/Zero-Based

Definition

Let $n \in \N$ be a natural number.

The initial segment of the natural numbers determined by $n$:

$\set {0, 1, 2, \ldots, n - 1}$

is denoted $\N_{<n}$.

Also denoted as

The usual notation for this is $\N_n$, but the notation $\N_{<n}$ is less ambiguous.

The notation $\Z_n$ is often see, for the set of least positive residues of the set of residue classes modulo $n$:

$\Z_n = \set {0, 1, \ldots, n - 1}$

which of course is exactly the same thing.

Also see

• Definition:Initial Segment of One-Based Natural Numbers

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers