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Secara analogi, diberikan bahwa penambahan himpunan bilangan asli didefinisikan di atas (lihat {{slink||Penambahan}}), operator [[perkalian]] <math>\times</math> dapat didefinisikan melalui <math> a \times 0 = 0 </math> dan <math> a \times S(b) = (a \times b) + a </math>. Ini mengubah <math> (\N^\star, \times) </math> menjadi monoid komutatif bebas dengan elemen identitas 1; generator set untuk monoid ini adalah himpunan [[bilangan prima]].
=== Hubungan antara penjumlahan dan perkalian ===
PenjumlahanPenambahan dan perkalian adalah kompatibel, yang dinyatakan dalam [[hukum distribusi|distribusi]]: {{math|''a'' × (''b'' + ''c'') {{=}} (''a'' × ''b'') + (''a'' × ''c'')}}. PropertiSifat penjumlahan dan perkalian ini membuat bilangan asli sebagai turunan dari [[komutatif]] [[semiring]]. SemiringsSemiring adalah generalisasi aljabar dari bilangan asli dimanadengan perkalian tidak seharusnya komutatif. Kurangnya aditif invers, yang setaraekuivalen dengan fakta bahwa {{<math|ℕ}}> \N </math> tidak [[penutupanKetertutupan (matematika)|tertutup]] dalamdi bawah pengurangan (yaitu, mengurangkan satu naturalbilangan asli dari bilangan asli yang lain tidak selalu menghasilkan naturalbilangan lainasli), berarti bahwa {{<math|ℕ}}> adalah\N </math> '' bukan bukanlah'' a [[gelanggang (matematika)|gelanggang]]; melainkan sebuah [[semiring]] (juga dikenal sebagai .
''gelanggang'')
Bila bilangan asli diambil sebagai "tidak termasuk 0", dan "mulai dari 1", definisi dari + dan × adalahdinyatakan seperti di atas, kecuali bahwa mereka diawali dengan {{math|''a'' + 1 {{=}} ''S''(''a'')}} and {{math|''a'' × 1 {{=}} ''a''}}.
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===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=2020-07-27 |website=mathworld.wolfram.com |lang=en}}</ref> and the standard [[order of operations]] is assumed.
=== Sifat aljabar yang dipenuhi bilangan asli===
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''}}.
Operasi penambahan (+) dan perkalian (×) pada bilangan asl, seperti yang didefinisikan sebelumnya, memiliki beberapa sifat-sifat aljabar:
* [[Ketertutupan (matematika)|Ketertutupan]] di bawah penambahan dan perkalian: untuk semua bilangan asli {{math|''a''}} dan {{math|''b''}}, maka {{math|''a'' + ''b''}} dan {{math|''a'' × ''b''}} adalah bilangan asli.<ref>{{cite book
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 [[omega (ordinal)|{{math|''ω''}}]] (omega).
| last1 = Fletcher | first1 = Harold
| last2 = Howell | first2 = Arnold A.
===Division===
| date = 2014-05-09
In this section, juxtaposed variables such as {{math|''ab''}} indicate the product {{math|''a'' × ''b''}}, and the standard [[order of operations]] is assumed.
| title = Mathematics with Understanding
| publisher = Elsevier
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 (mathematics)|division]] with remainder'' 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
| isbn = 978-1-4832-8079-0
:{{math|''a'' {{=}} ''bq'' + ''r''}} and {{math|''r'' < ''b''}}.
| page = 116
| lang = en
The number {{math|''q''}} is called the ''[[quotient]]'' and {{math|''r''}} is called the ''[[remainder]]'' of the division of {{math|''a''}} by {{math|''b''}}. The numbers {{math|''q''}} and {{math|''r''}} are uniquely determined by {{math|''a''}} and {{math|''b''}}. This [[Euclidean division]] is key to several other properties ([[divisibility]]), algorithms (such as the [[Euclidean algorithm]]), and ideas in number theory.
| url = https://books.google.com/books?id=7cPSBQAAQBAJ&pg=PA116
| quote = ...the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication}}</ref>
===Algebraic properties satisfied by the natural numbers===
* [[ DistributivitySifat asosiatif|Pengelompokan]] : of multiplication over addition foruntuk allsemua naturalbilangan numbersasli {{math|''a''}}, {{math|''b''}}, anddan {{math|''c''}}, maka {{math|''a'' ×+ (''b'' + ''c'') {{=}} (''a'' ×+ ''b'') + (''c''}} dan {{math|''a '' × (''b'' × ''c'') {{=}} (''a'' × ''b'') × ''c''}}. <ref>{{cite book▼The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties:
| last = Davisson | first = Schuyler Colfax
* [[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=2014-05-09 |title=Mathematics with Understanding |publisher=Elsevier |isbn=978-1-4832-8079-0 |page=116 |lang=en |url=https://books.google.com/books?id=7cPSBQAAQBAJ&newbks=0&printsec=frontcover&pg=PA116&dq=Natural+numbers+closed&hl=en |quote=...the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication}}</ref>
| title = College Algebra
* [[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 |lang=en |url=https://books.google.com/books?id=E7oZAAAAYAAJ&newbks=0&printsec=frontcover&pg=PA2&dq=Natural+numbers+associative&hl=en |quote=Addition of natural numbers is associative.}}</ref>
| date = 1910
* [[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 |lang=en |url=https://books.google.com/books?id=xERMAQAAIAAJ&newbks=0&printsec=frontcover&dq=Natural+numbers+commutative&q=Natural+numbers+commutative&hl=en}}</ref>
| publisher = Macmillian Company
* Existence of [[identity element]]s: for every natural number ''a'', {{math|''a'' + 0 {{=}} ''a''}} and {{math|''a'' × 1 {{=}} ''a''}}. ▼ | page = 2
▲* [[Distributivity]] of multiplication over addition for all natural numbers {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}, {{math|''a'' × (''b'' + ''c'') {{=}} (''a'' × ''b'') + (''a'' × ''c'')}}.
| lang = en
* 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).--> ▼ | url = https://books.google.com/books?id=E7oZAAAAYAAJ&pg=PA2
| quote = Addition of natural numbers is associative.}}</ref>
* [[Sifat komutatif|Pertukaran]]: untuk semu bilangan asli {{math|''a''}} dan {{math|''b''}}, maka {{math|''a'' + ''b'' {{=}} ''b'' + ''a''}} dan {{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
| lang = en
| url = https://books.google.com/books?id=xERMAQAAIAAJ&newbks=0&printsec=frontcover&dq=Natural+numbers+commutative&q=Natural+numbers+commutative&hl=en}}</ref>
▲* Existence ofKeberadaan [[ identityelemen elementidentitas]] s: foruntuk everysetiap naturalbilangan numberasli {{math|''a'' }}, {{math|''a'' + 0 {{=}} ''a''}} anddan {{math|''a'' × 1 {{=}} ''a''}}.
* [[Sifat distributif|Distribusi]] dari perkalian atas penambahan untuk semua bilangan asli {{math|''a''}}, {{math|''b''}}, dan {{math|''c''}}, {{math|''a'' × (''b'' + ''c'') {{=}} (''a'' × ''b'') + (''a'' × ''c'')}}.
▲* NoTidak nonzeroada [[ zeropembagi divisornol]] s tak-nol: ifbila {{math|''a''}} anddan {{math|''b''}} areadalah naturalbilangan numbersasli such thatsehingga {{math|''a'' × ''b'' {{=}} 0}}, thenmaka {{math|''a'' {{=}} 0}} oratau {{math|''b'' {{=}} 0}} ( oratau bothkedua-duanya). -->
=== Ketakhinggaan ===
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