# Definition:Initial Segment of Natural Numbers/Zero-Based

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## 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

## Sources

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