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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?

  • $\mathbb{N}_0$
  • $\mathbb{N}\cup\{0\}$
  • $\mathbb{Z}_{\ge 0}$
  • $\mathbb{Z}_{+}$
  • $\mathbb{Z}_{0+}$
  • $\mathbb{Z}_{*}$
  • $\mathbb{Z}_{\geq}$
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    $\begingroup$ They all seem clear enough to me, except maybe $\mathbb{Z}_+$, which might not include $0$ :/ $\endgroup$
    – G Tony Jacobs
    Mar 12, 2014 at 18:37
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    $\begingroup$ 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. $\endgroup$
    – user133281
    Mar 12, 2014 at 18:39
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    $\begingroup$ $\mathbb{Z}_+$ 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. $\endgroup$
    – Daniel Fischer
    Mar 12, 2014 at 18:39
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    $\begingroup$ @DanielFischer. Some people use the definition that $0\notin \mathbb{N}$. Hence, $\mathbb{N}$ alone is ambiguous. $\endgroup$
    – Batominovski
    Aug 19, 2015 at 2:06
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    $\begingroup$ You forgot $\omega$! $\endgroup$
    – user29123
    Aug 19, 2015 at 2:35

5 Answers 5

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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 $$ \mathbb{N}^* = \mathbb{N}^+ = \mathbb{N}_1 = \mathbb{N}_{>0}= \{ 1, 2, \ldots \}. $$

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

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

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

Nonnegative integer

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    $\begingroup$ I might interpret that as either the nonzero integers or as the group of units of the integers. $\endgroup$
    – Qiaochu Yuan
    Oct 19, 2015 at 22:25
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    $\begingroup$ Right, those are the things I would interpret $\mathbb{Z}^{\ast}$ as. Very confusing notation on Mathworld. $\endgroup$
    – Daniel Fischer
    Oct 19, 2015 at 22:26
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    $\begingroup$ 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. $\endgroup$
    – Wood
    Aug 7, 2016 at 4:55
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The set of numbers $\{0, 1, 2, \dots\}$ is well-known as the set of whole numbers $\mathbb{W}$.

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  • $\begingroup$ Interesting. Could you provide a citation? $\endgroup$
    – jasonseminara
    Jul 17, 2021 at 3:44
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I personally always use $\Bbb N_0$ because what you are really describing is just the natural numbers plus the element $\{0\}$.

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