# The best symbol for non-negative integers?

I would like to specify the set $$\{0, 1, 2, \dots\}$$, i.e., non-negative integers in an engineering conference paper. Which symbol is more preferable?

• $$\mathbb{N}_0$$
• $$\mathbb{N}\cup\{0\}$$
• $$\mathbb{Z}_{\ge 0}$$
• $$\mathbb{Z}_{+}$$
• $$\mathbb{Z}_{0+}$$
• $$\mathbb{Z}_{*}$$
• $$\mathbb{Z}_{\geq}$$
• They all seem clear enough to me, except maybe $\mathbb{Z}_+$, which might not include $0$ :/ Mar 12, 2014 at 18:37
• In my opinion, a notation using $\mathbb{Z}$ (such as $\mathbb{Z}_{\geq 0}$) is preferable over a notation using $\mathbb{N}$, a symbol that means different things in different countries. Mar 12, 2014 at 18:39
• $\mathbb{Z}_+$ looks like the set of strictly positive integers to me. $\mathbb{N}\cup \{0\}$ is unambiguous, even if it is redundant ('cause, you know, $0\in\mathbb{N}$). $\mathbb{Z}_{\geqslant 0}$ is also clear. Mar 12, 2014 at 18:39
• @DanielFischer. Some people use the definition that $0\notin \mathbb{N}$. Hence, $\mathbb{N}$ alone is ambiguous. Aug 19, 2015 at 2:06
• You forgot $\omega$!
– user29123
Aug 19, 2015 at 2:35

According to Wikipedia, unambiguous notations for the set of non-negative integers include $$\mathbb{N}^0 = \mathbb{N}_0 = \{ 0, 1, 2, \ldots \},$$ while the set of positive integers may be denoted unambiguously by $$\mathbb{N}^* = \mathbb{N}^+ = \mathbb{N}_1 = \mathbb{N}_{>0}= \{ 1, 2, \ldots \}.$$

Based on this similar post, the following seems to be preferred:

$\mathbb{Z}_{\geq 0}$

Wolfram Mathworld has $\mathbb{Z}^*$.

Nonnegative integer

• I might interpret that as either the nonzero integers or as the group of units of the integers. Oct 19, 2015 at 22:25
• Right, those are the things I would interpret $\mathbb{Z}^{\ast}$ as. Very confusing notation on Mathworld. Oct 19, 2015 at 22:26
• In my opinion, it's a bad notation; but this answer is valid, since it references Wolfram Mathworld, which is a popular and reliable source.
– Wood
Aug 7, 2016 at 4:55

The set of numbers $$\{0, 1, 2, \dots\}$$ is well-known as the set of whole numbers $$\mathbb{W}$$.

• Interesting. Could you provide a citation? Jul 17, 2021 at 3:44

I personally always use $$\Bbb N_0$$ because what you are really describing is just the natural numbers plus the element $$\{0\}$$.