Editing Natural number (section)

==Notation==
The [[Set (mathematics)|set]] of all natural numbers is standardly denoted {{math|'''N'''}} or <math>\mathbb N.</math><ref name=":1"/><ref>{{cite web |title=Listing of the Mathematical Notations used in the Mathematical Functions Website: Numbers, variables, and functions |url=https://functions.wolfram.com/Notations/1/ |access-date=27 July 2020 |website=functions.wolfram.com}}</ref> Older texts have occasionally employed {{math|''J''}} as the symbol for this set.<ref>{{cite book |url=https://archive.org/details/1979RudinW |title=Principles of Mathematical Analysis |last=Rudin |first=W. |publisher=McGraw-Hill |year=1976 |isbn=978-0-07-054235-8 |location=New York |page=25}}</ref>

Since natural numbers may contain {{math|0}} or not, it may be important to know which version is referred to. This is often specified by the context, but may also be done by using a subscript or a superscript in the notation, such as:<ref name="ISO80000"/><ref name="Grimaldi">{{cite book |last1=Grimaldi |first1=Ralph P. |title=Discrete and Combinatorial Mathematics: An applied introduction |publisher=Pearson Addison Wesley |isbn=978-0-201-72634-3 |edition=5th |year=2004}}</ref>
* Naturals without zero: <math>\{1,2,...\}=\mathbb{N}^*= \mathbb N^+=\mathbb{N}_0\smallsetminus\{0\} = \mathbb{N}_1</math>
* Naturals with zero: <math>\;\{0,1,2,...\}=\mathbb{N}_0=\mathbb N^0=\mathbb{N}^*\cup\{0\}</math>

Alternatively, since the natural numbers naturally form a [[subset]] of the [[integer]]s (often {{nowrap|denoted <math>\mathbb Z</math>),}} they may be referred to as the positive, or the non-negative integers, respectively.<ref>{{cite book |last1=Grimaldi |first1=Ralph P. |title=A review of discrete and combinatorial mathematics |date=2003 |publisher=Addison-Wesley |location=Boston |isbn=978-0-201-72634-3 |page=133 |edition=5th}}</ref> To be unambiguous about whether 0 is included or not, sometimes a superscript "<math>*</math>" or "+" is added in the former case, and a subscript (or superscript) "0" is added in the latter case:<ref name=ISO80000 >{{cite book |title=ISO 80000-2:2019 |chapter-url=https://cdn.standards.iteh.ai/samples/64973/329519100abd447ea0d49747258d1094/ISO-80000-2-2019.pdf#page=10 |publisher=[[International Organization for Standardization]]| chapter = Standard number sets and intervals | date=19 May 2020 |page=4|url=https://www.iso.org/standard/64973.html}}</ref>

:<math>\{1, 2, 3,\dots\} = \{x \in \mathbb Z : x > 0\}=\mathbb Z^+= \mathbb{Z}_{>0}</math>
:<math>\{0, 1, 2,\dots\} = \{x \in \mathbb Z : x \ge 0\}=\mathbb Z^{+}_{0}=\mathbb{Z}_ {\ge 0}</math>