Editing Natural number (section)

==Properties==
This section uses the convention <math>\mathbb{N}=\mathbb{N}_0=\mathbb{N}^*\cup\{0\}</math>.

===Addition===
Given the set <math>\mathbb{N}</math> of natural numbers and the [[successor function]] <math>S \colon \mathbb{N} \to \mathbb{N}</math> sending each natural number to the next one, one can define [[Addition in N|addition]] of natural numbers recursively by setting {{math|''a'' + 0 {{=}} ''a''}} and {{math|''a'' + ''S''(''b'') {{=}} ''S''(''a'' + ''b'')}} for all {{math|''a''}}, {{math|''b''}}. Thus, {{math|''a'' + 1 {{=}} ''a'' + S(0) {{=}} S(''a''+0) {{=}} S(''a'')}}, {{math|''a'' + 2 {{=}} ''a'' + S(1) {{=}} S(''a''+1) {{=}} S(S(''a''))}}, and so on. The [[algebraic structure]] <math>(\mathbb{N}, +)</math> is a [[commutative]] [[monoid]] with [[identity element]]&nbsp;0. It is a [[free object|free monoid]] on one generator. This commutative monoid satisfies the [[cancellation property]], so it can be embedded in a [[group (mathematics)|group]]. The smallest group containing the natural numbers is the [[integer]]s.

If 1 is defined as {{math|''S''(0)}}, then {{math|''b'' + 1 {{=}} ''b'' + ''S''(0) {{=}} ''S''(''b'' + 0) {{=}} ''S''(''b'')}}. That is, {{math|''b'' + 1}} is simply the successor of {{math|''b''}}.

===Multiplication===
Analogously, given that addition has been defined, a [[multiplication]] operator <math>\times</math> can be defined via {{math|''a'' × 0 {{=}} 0}} and {{math|''a'' × S(''b'') {{=}} (''a'' × ''b'') + ''a''}}. This turns <math>(\mathbb{N}^*, \times)</math> into a [[free commutative monoid]] with identity element&nbsp;1; a generator set for this monoid is the set of [[prime number]]s.

===Relationship between addition and multiplication===
Addition and multiplication are compatible, which is expressed in the [[distributivity|distribution law]]: {{math|''a'' × (''b'' + ''c'') {{=}} (''a'' × ''b'') + (''a'' × ''c'')}}. These properties of addition and multiplication make the natural numbers an instance of a [[commutative]] [[semiring]]. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that <math>\mathbb{N}</math> is not [[closure (mathematics)|closed]] under subtraction (that is, subtracting one natural from another does not always result in another natural), means that <math>\mathbb{N}</math> is ''not'' a [[ring (mathematics)|ring]]; instead it is a [[semiring]] (also known as a ''rig'').

If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin with {{math|''a'' + 1 {{=}} ''S''(''a'')}} and {{math|''a'' × 1 {{=}} ''a''}}. Furthermore, <math>(\mathbb{N^*}, +)</math> has no identity element.

===Order===
In this section, juxtaposed variables such as {{math|''ab''}} indicate the product {{math|''a'' × ''b''}},<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Multiplication |url=https://mathworld.wolfram.com/Multiplication.html |access-date=27 July 2020 |website=mathworld.wolfram.com |language=en}}</ref> and the standard [[order of operations]] is assumed.

A [[total order]] on the natural numbers is defined by letting {{math|''a'' ≤ ''b''}} if and only if there exists another natural number {{math|''c''}} where {{math|''a'' + ''c'' {{=}} ''b''}}. This order is compatible with the [[arithmetical operations]] in the following sense: if {{math|''a''}}, {{math|''b''}} and {{math|''c''}} are natural numbers and {{math|''a'' ≤ ''b''}}, then {{math|''a'' + ''c'' ≤ ''b'' + ''c''}} and {{math|''ac'' ≤ ''bc''}}.

An important property of the natural numbers is that they are [[well-order]]ed: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an [[ordinal number]]; for the natural numbers, this is denoted as {{math|[[omega (ordinal)|''ω'']]}} (omega).

===Division===
In this section, juxtaposed variables such as {{math|''ab''}} indicate the product {{math|''a'' × ''b''}}, and the standard [[order of operations]] is assumed.

While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of ''division with remainder'' or [[Euclidean division]] is available as a substitute: for any two natural numbers {{math|''a''}} and {{math|''b''}} with {{math|''b'' ≠ 0}} there are natural numbers {{math|''q''}} and {{math|''r''}} such that
:<math>a = bq + r \text{ and } r < b. </math>

The number {{math|''q''}} is called the ''[[quotient]]'' and {{math|''r''}} is called the ''[[remainder]]'' of the division of {{math|''a''}} by&nbsp;{{math|''b''}}. The numbers {{math|''q''}} and {{math|''r''}} are uniquely determined by {{math|''a''}} and&nbsp;{{math|''b''}}. This Euclidean division is key to the several other properties ([[divisibility]]), algorithms (such as the [[Euclidean algorithm]]), and ideas in number theory.

===Algebraic properties satisfied by the natural numbers===
The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties:
* [[Closure (mathematics)|Closure]] under addition and multiplication: for all natural numbers {{math|''a''}} and {{math|''b''}}, both {{math|''a'' + ''b''}} and {{math|''a'' × ''b''}} are natural numbers.<ref>{{cite book |last1=Fletcher |first1=Harold |last2=Howell |first2=Arnold A. |date=9 May 2014 |title=Mathematics with Understanding |publisher=Elsevier |isbn=978-1-4832-8079-0 |page=116 |language=en |url=https://books.google.com/books?id=7cPSBQAAQBAJ&q=Natural+numbers+closed&pg=PA116 |quote=...the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication}}</ref>
* [[Associativity]]: for all natural numbers {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}, {{math|''a'' + (''b'' + ''c'') {{=}} (''a'' + ''b'') + ''c''}} and {{math|''a'' × (''b'' × ''c'') {{=}} (''a'' × ''b'') × ''c''}}.<ref>{{cite book |last=Davisson |first=Schuyler Colfax |title=College Algebra |date=1910 |publisher=Macmillian Company |page=2 |language=en |url=https://books.google.com/books?id=E7oZAAAAYAAJ&q=Natural+numbers+associative&pg=PA2 |quote=Addition of natural numbers is associative.}}</ref>
* [[Commutativity]]: for all natural numbers {{math|''a''}} and {{math|''b''}}, {{math|''a'' + ''b'' {{=}} ''b'' + ''a''}} and {{math|''a'' × ''b'' {{=}} ''b'' × ''a''}}.<ref>{{cite book |last1=Brandon |first1=Bertha (M.) |last2=Brown |first2=Kenneth E. |last3=Gundlach |first3=Bernard H. |last4=Cooke |first4=Ralph J. |date=1962 |title=Laidlaw mathematics series |publisher=Laidlaw Bros. |volume=8 |page=25 |language=en |url=https://books.google.com/books?id=xERMAQAAIAAJ&q=Natural+numbers+commutative}}</ref>
* Existence of [[identity element]]s: for every natural number {{Math|''a''}}, {{math|''a'' + 0 {{=}} ''a''}} and {{math|''a'' × 1 {{=}} ''a''}}. 
** If the natural numbers are taken as "excluding 0", and "starting at 1", then for every natural number {{Math|''a''}}, {{math|''a'' × 1 {{=}} ''a''}}. However, the "existence of additive identity element" property is not satisfied
* [[Distributivity]] of multiplication over addition for all natural numbers {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}, {{math|''a'' × (''b'' + ''c'') {{=}} (''a'' × ''b'') + (''a'' × ''c'')}}.
* No nonzero [[zero divisor]]s: if {{math|''a''}} and {{math|''b''}} are natural numbers such that {{math|''a'' × ''b'' {{=}} 0}}, then {{math|''a'' {{=}} 0}} or {{math|''b'' {{=}} 0}} (or both).