Editing Natural number (section)

===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).