Symbols:N

The Set of Natural Numbers

 * $\N$

The set of natural numbers:
 * $\N = \left\{{0, 1, 2, 3, \ldots}\right\}$.

The Set of Non-Zero Natural Numbers

 * $\N_{> 0}$ or $\N_{\ne 0}$

The set of non-zero natural numbers:
 * $\N_{> 0} = \left\{{1, 2, 3, \ldots}\right\}$.

Deprecated

 * $\N^*$

The set of non-zero natural numbers:
 * $\N^* = \left\{{1, 2, 3, \ldots}\right\}$.

Initial Segment of Natural Numbers

 * $\N_n$, $\N^*_n$

The set $\N_n$ is the set of all natural numbers which are less than $n$:
 * $\N_n = \left\{{x \in \N: x < n}\right\} = \left\{{0, 1, 2, \ldots, n-1}\right\}$.

Similarly, the set $\N^*_n$ is the set of all non-zero natural numbers which are less or equal to $n$:
 * $\N^*_n = \left\{{x \in \N^*: x \le n}\right\} = \left\{{1, 2, \ldots, n}\right\}$.

= Deprecated Usages =

Older literature tends to use $\N$ to mean $\left\{{1, 2, 3, \ldots}\right\}$.

Consequently, the set $\left\{{0, 1, 2, 3, \ldots}\right\}$ needs another symbol to denote it. The usual technique is to use $\tilde {\N}$.