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class="banner-container"> <div id="siteNotice"></div> </div> <div class="pre-content heading-holder"> <div class="page-heading"> <h1 id="firstHeading" class="firstHeading mw-first-heading">Natural number</h1> <div class="tagline"></div> </div> <ul id="p-associated-pages" class="minerva__tab-container"> <li class="minerva__tab selected"> <a class="minerva__tab-text" href="/wiki/Natural_number" rel="" data-event-name="tabs.subject">Article</a> </li> <li class="minerva__tab "> <a class="minerva__tab-text" href="/wiki/Talk:Natural_number" rel="discussion" data-event-name="tabs.talk">Talk</a> </li> </ul> <nav class="page-actions-menu"> <ul id="p-views" class="page-actions-menu__list"> <li id="language-selector" class="page-actions-menu__list-item"> <a role="button" href="#p-lang" data-mw="interface" data-event-name="menu.languages" title="Language" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet language-selector"> Language </a> </li> <li id="page-actions-watch" class="page-actions-menu__list-item"> <a role="button" id="ca-watch" href="/w/index.php?title=Special:UserLogin&returnto=Natural+number" data-event-name="menu.watch" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet menu__item--page-actions-watch"> Watch </a> </li> <li id="page-actions-edit" class="page-actions-menu__list-item"> <a role="button" id="ca-edit" href="/w/index.php?title=Natural_number&action=edit" data-event-name="menu.edit" data-mw="interface" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> Edit </a> </li> </ul> </nav>  <div id="mw-content-subtitle">(Redirected from <a href="/w/index.php?title=Natural_numbers&redirect=no" class="mw-redirect" title="Natural numbers">Natural numbers</a>)</div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"><script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><section class="mf-section-0" id="mf-section-0"> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about "positive integers" and "non-negative integers". For all the numbers ..., −2, −1, 0, 1, 2, ..., see <a href="/wiki/Integer" title="Integer">Integer</a>.</div> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the natural numbers are the <a href="/wiki/Number" title="Number">numbers</a> 0, 1, 2, 3, and so on, possibly excluding 0.<a href="#cite_note-Enderton-1">[1]</a> Some start counting with 0, defining the natural numbers as the non-negative integers 0, 1, 2, 3, ..., while others start with 1, defining them as the positive integers 1, 2, 3, ... .<a href="#cite_note-2">[a]</a> Some authors acknowledge both definitions whenever convenient.<a href="#cite_note-:1-3">[2]</a> Sometimes, the whole numbers are the natural numbers plus zero. In other cases, the whole numbers refer to all of the <a href="/wiki/Integer" title="Integer">integers</a>, including negative integers.<a href="#cite_note-4">[3]</a> The counting numbers are another term for the natural numbers, particularly in primary school education, and are ambiguous as well although typically start at 1.<a href="#cite_note-MathWorld_CountingNumber-5">[4]</a> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Three_Baskets_with_Apples.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Three_Baskets_with_Apples.svg/170px-Three_Baskets_with_Apples.svg.png" decoding="async" width="170" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Three_Baskets_with_Apples.svg/255px-Three_Baskets_with_Apples.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/Three_Baskets_with_Apples.svg/340px-Three_Baskets_with_Apples.svg.png 2x" data-file-width="298" data-file-height="274"></a><figcaption>Natural numbers can be used for counting: one apple; two apples are one apple added to another apple, three apples are one apple added to two apples, ...</figcaption></figure> The natural numbers are used for counting things, like "there are six coins on the table", in which case they are called <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal numbers</a>. They are also used to put things in order, like "this is the third largest city in the country", which are called <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal numbers</a>. Natural numbers are also used as labels, like <a href="/wiki/Number_(sports)" title="Number (sports)">jersey numbers</a> on a sports team, where they serve as <a href="/wiki/Nominal_number" title="Nominal number">nominal numbers</a> and do not have mathematical properties.<a href="#cite_note-6">[5]</a> The natural numbers form a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>, commonly symbolized as a bold N or <a href="/wiki/Blackboard_bold" title="Blackboard bold">blackboard bold</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }">⁠. Many other <a href="/wiki/Number_set" class="mw-redirect" title="Number set">number sets</a> are built from the natural numbers. For example, the <a href="/wiki/Integer" title="Integer">integers</a> are made by adding 0 and negative numbers. The <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> add fractions, and the <a href="/wiki/Real_number" title="Real number">real numbers</a> add infinite decimals. <a href="/wiki/Complex_number" title="Complex number">Complex numbers</a> add the <a href="/wiki/Imaginary_unit" title="Imaginary unit">square root of −1</a>. This chain of extensions canonically <a href="/wiki/Embedding" title="Embedding">embeds</a> the natural numbers in the other number systems.<a href="#cite_note-7">[6]</a><a href="#cite_note-8">[7]</a> Natural numbers are studied in different areas of math. <a href="/wiki/Number_theory" title="Number theory">Number theory</a> looks at things like how numbers divide evenly (<a href="/wiki/Divisibility" class="mw-redirect" title="Divisibility">divisibility</a>), or how <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> are spread out. <a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a> studies counting and arranging numbered objects, such as <a href="/wiki/Partition_(number_theory)" class="mw-redirect" title="Partition (number theory)">partitions</a> and <a href="/wiki/Enumerative_combinatorics" title="Enumerative combinatorics">enumerations</a>. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#History">1 History</a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Ancient_roots">1.1 Ancient roots</a></li> <li class="toclevel-2 tocsection-3"><a href="#Emergence_as_a_term">1.2 Emergence as a term</a></li> <li class="toclevel-2 tocsection-4"><a href="#Formal_construction">1.3 Formal construction</a></li> </ul> </li> <li class="toclevel-1 tocsection-5"><a href="#Notation">2 Notation</a></li> <li class="toclevel-1 tocsection-6"><a href="#Properties">3 Properties</a> <ul> <li class="toclevel-2 tocsection-7"><a href="#Addition">3.1 Addition</a></li> <li class="toclevel-2 tocsection-8"><a href="#Multiplication">3.2 Multiplication</a></li> <li class="toclevel-2 tocsection-9"><a href="#Relationship_between_addition_and_multiplication">3.3 Relationship between addition and multiplication</a></li> <li class="toclevel-2 tocsection-10"><a href="#Order">3.4 Order</a></li> <li class="toclevel-2 tocsection-11"><a href="#Division">3.5 Division</a></li> <li class="toclevel-2 tocsection-12"><a href="#Algebraic_properties_satisfied_by_the_natural_numbers">3.6 Algebraic properties satisfied by the natural numbers</a></li> </ul> </li> <li class="toclevel-1 tocsection-13"><a href="#Generalizations">4 Generalizations</a></li> <li class="toclevel-1 tocsection-14"><a href="#Formal_definitions">5 Formal definitions</a> <ul> <li class="toclevel-2 tocsection-15"><a href="#Peano_axioms">5.1 Peano axioms</a></li> <li class="toclevel-2 tocsection-16"><a href="#Set-theoretic_definition">5.2 Set-theoretic definition</a></li> </ul> </li> <li class="toclevel-1 tocsection-17"><a href="#See_also">6 See also</a></li> <li class="toclevel-1 tocsection-18"><a href="#Notes">7 Notes</a></li> <li class="toclevel-1 tocsection-19"><a href="#References">8 References</a></li> <li class="toclevel-1 tocsection-20"><a href="#Bibliography">9 Bibliography</a></li> <li class="toclevel-1 tocsection-21"><a href="#External_links">10 External links</a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><h2 id="History">History</h2> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=1" title="Edit section: History" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> <div class="mw-heading mw-heading3"><h3 id="Ancient_roots">Ancient roots</h3> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=2" title="Edit section: Ancient roots" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Prehistoric_counting" title="Prehistoric counting">Prehistoric counting</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ishango_bone_(cropped).jpg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Ishango_bone_%28cropped%29.jpg/220px-Ishango_bone_%28cropped%29.jpg" decoding="async" width="220" height="304" class="mw-file-element" data-file-width="891" data-file-height="1230"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 304px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Ishango_bone_%28cropped%29.jpg/220px-Ishango_bone_%28cropped%29.jpg" data-width="220" data-height="304" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Ishango_bone_%28cropped%29.jpg/330px-Ishango_bone_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Ishango_bone_%28cropped%29.jpg/440px-Ishango_bone_%28cropped%29.jpg 2x" data-class="mw-file-element"> </a><figcaption>The <a href="/wiki/Ishango_bone" title="Ishango bone">Ishango bone</a> (on exhibition at the <a href="/wiki/Royal_Belgian_Institute_of_Natural_Sciences" class="mw-redirect" title="Royal Belgian Institute of Natural Sciences">Royal Belgian Institute of Natural Sciences</a>)<a href="#cite_note-RBINS_intro-9">[8]</a><a href="#cite_note-RBINS_flash-10">[9]</a><a href="#cite_note-UNESCO-11">[10]</a> is believed to have been used 20,000 years ago for natural number arithmetic.</figcaption></figure> The most primitive method of representing a natural number is to use one's fingers, as in <a href="/wiki/Finger_counting" class="mw-redirect" title="Finger counting">finger counting</a>. Putting down a <a href="/wiki/Tally_mark" class="mw-redirect" title="Tally mark">tally mark</a> for each object is another primitive method. Later, a set of objects could be tested for equality, excess or shortage—by striking out a mark and removing an object from the set. The first major advance in abstraction was the use of <a href="/wiki/Numeral_system" title="Numeral system">numerals</a> to represent numbers. This allowed systems to be developed for recording large numbers. The ancient <a href="/wiki/History_of_Ancient_Egypt" class="mw-redirect" title="History of Ancient Egypt">Egyptians</a> developed a powerful system of numerals with distinct <a href="/wiki/Egyptian_hieroglyphs" title="Egyptian hieroglyphs">hieroglyphs</a> for 1, 10, and all powers of 10 up to over 1 million. A stone carving from <a href="/wiki/Karnak" title="Karnak">Karnak</a>, dating back from around 1500 BCE and now at the <a href="/wiki/Louvre" title="Louvre">Louvre</a> in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The <a href="/wiki/Babylonia" title="Babylonia">Babylonians</a> had a <a href="/wiki/Positional_notation" title="Positional notation">place-value</a> system based essentially on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one—its value being determined from context.<a href="#cite_note-12">[11]</a> A much later advance was the development of the idea that <a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">0</a> can be considered as a number, with its own numeral. The use of a 0 <a href="/wiki/Numerical_digit" title="Numerical digit">digit</a> in place-value notation (within other numbers) dates back as early as 700 BCE by the Babylonians, who omitted such a digit when it would have been the last symbol in the number.<a href="#cite_note-14">[b]</a> The <a href="/wiki/Olmec" class="mw-redirect" title="Olmec">Olmec</a> and <a href="/wiki/Maya_civilization" title="Maya civilization">Maya civilizations</a> used 0 as a separate number as early as the 1st century BCE, but this usage did not spread beyond <a href="/wiki/Mesoamerica" title="Mesoamerica">Mesoamerica</a>.<a href="#cite_note-15">[13]</a><a href="#cite_note-16">[14]</a> The use of a numeral 0 in modern times originated with the Indian mathematician <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a> in 628 CE. However, 0 had been used as a number in the medieval <a href="/wiki/Computus" class="mw-redirect" title="Computus">computus</a> (the calculation of the date of Easter), beginning with <a href="/wiki/Dionysius_Exiguus" title="Dionysius Exiguus">Dionysius Exiguus</a> in 525 CE, without being denoted by a numeral. Standard <a href="/wiki/Roman_numerals" title="Roman numerals">Roman numerals</a> do not have a symbol for 0; instead, nulla (or the genitive form nullae) from nullus, the Latin word for "none", was employed to denote a 0 value.<a href="#cite_note-17">[15]</a> The first systematic study of numbers as <a href="/wiki/Abstraction" title="Abstraction">abstractions</a> is usually credited to the <a href="/wiki/Ancient_Greece" title="Ancient Greece">Greek</a> philosophers <a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a> and <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a>. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.<a href="#cite_note-19">[c]</a> <a href="/wiki/Euclid" title="Euclid">Euclid</a>, for example, defined a unit first and then a number as a multitude of units, thus by his definition, a unit is not a number and there are no unique numbers (e.g., any two units from indefinitely many units is a 2).<a href="#cite_note-Mueller_2006_p._58-20">[17]</a> However, in the definition of <a href="/wiki/Perfect_number" title="Perfect number">perfect number</a> which comes shortly afterward, Euclid treats 1 as a number like any other.<a href="#cite_note-21">[18]</a> Independent studies on numbers also occurred at around the same time in <a href="/wiki/India" title="India">India</a>, China, and <a href="/wiki/Mesoamerica" title="Mesoamerica">Mesoamerica</a>.<a href="#cite_note-22">[19]</a> <div class="mw-heading mw-heading3"><h3 id="Emergence_as_a_term">Emergence as a term</h3> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=3" title="Edit section: Emergence as a term" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <a href="/wiki/Nicolas_Chuquet" title="Nicolas Chuquet">Nicolas Chuquet</a> used the term progression naturelle (natural progression) in 1484.<a href="#cite_note-23">[20]</a> The earliest known use of "natural number" as a complete English phrase is in 1763.<a href="#cite_note-24">[21]</a><a href="#cite_note-MacTutor-25">[22]</a> The 1771 Encyclopaedia Britannica defines natural numbers in the logarithm article.<a href="#cite_note-MacTutor-25">[22]</a> Starting at 0 or 1 has long been a matter of definition. In 1727, <a href="/wiki/Bernard_Le_Bovier_de_Fontenelle" title="Bernard Le Bovier de Fontenelle">Bernard Le Bovier de Fontenelle</a> wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0.<a href="#cite_note-26">[23]</a> In 1889, <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a> used N for the positive integers and started at 1,<a href="#cite_note-27">[24]</a> but he later changed to using N0 and N1.<a href="#cite_note-28">[25]</a> Historically, most definitions have excluded 0,<a href="#cite_note-MacTutor-25">[22]</a><a href="#cite_note-29">[26]</a><a href="#cite_note-30">[27]</a> but many mathematicians such as <a href="/wiki/George_A._Wentworth" title="George A. Wentworth">George A. Wentworth</a>, <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>, <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Nicolas Bourbaki</a>, <a href="/wiki/Paul_Halmos" title="Paul Halmos">Paul Halmos</a>, <a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Stephen Cole Kleene</a>, and <a href="/wiki/John_Horton_Conway" title="John Horton Conway">John Horton Conway</a> have preferred to include 0.<a href="#cite_note-31">[28]</a><a href="#cite_note-MacTutor-25">[22]</a> Mathematicians have noted tendencies in which definition is used, such as algebra texts including 0,<a href="#cite_note-MacTutor-25">[22]</a><a href="#cite_note-MacLaneBirkhoff1999p15-32">[d]</a> number theory and analysis texts excluding 0,<a href="#cite_note-MacTutor-25">[22]</a><a href="#cite_note-K%C5%99%C3%AD%C5%BEek-33">[29]</a><a href="#cite_note-34">[30]</a> logic and set theory texts including 0,<a href="#cite_note-35">[31]</a><a href="#cite_note-36">[32]</a><a href="#cite_note-37">[33]</a> dictionaries excluding 0,<a href="#cite_note-MacTutor-25">[22]</a><a href="#cite_note-38">[34]</a> school books (through high-school level) excluding 0, and upper-division college-level books including 0.<a href="#cite_note-Enderton-1">[1]</a> There are exceptions to each of these tendencies and as of 2023 no formal survey has been conducted. Arguments raised include <a href="/wiki/Division_by_zero" title="Division by zero">division by zero</a><a href="#cite_note-K%C5%99%C3%AD%C5%BEek-33">[29]</a> and the size of the <a href="/wiki/Empty_set" title="Empty set">empty set</a>. <a href="/wiki/Computer_language" title="Computer language">Computer languages</a> often <a href="/wiki/Zero-based_numbering" title="Zero-based numbering">start from zero</a> when enumerating items like <a href="/wiki/For_loop" title="For loop">loop counters</a> and <a href="/wiki/String_(computer_science)" title="String (computer science)">string-</a> or <a href="/wiki/Array_data_structure" class="mw-redirect" title="Array data structure">array-elements</a>.<a href="#cite_note-39">[35]</a><a href="#cite_note-40">[36]</a> Including 0 began to rise in popularity in the 1960s.<a href="#cite_note-MacTutor-25">[22]</a> The <a href="/wiki/ISO_31-11" title="ISO 31-11">ISO 31-11</a> standard included 0 in the natural numbers in its first edition in 1978 and this has continued through its present edition as <a href="/wiki/ISO/IEC_80000" title="ISO/IEC 80000">ISO 80000-2</a>.<a href="#cite_note-ISO80000-41">[37]</a> <div class="mw-heading mw-heading3"><h3 id="Formal_construction">Formal construction</h3> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=4" title="Edit section: Formal construction" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> stated that axioms can only be demonstrated in their finite application, and concluded that it is "the power of the mind" which allows conceiving of the indefinite repetition of the same act.<a href="#cite_note-42">[38]</a> <a href="/wiki/Leopold_Kronecker" title="Leopold Kronecker">Leopold Kronecker</a> summarized his belief as "God made the integers, all else is the work of man".<a href="#cite_note-45">[e]</a> The <a href="/wiki/Constructivism_(mathematics)" class="mw-redirect" title="Constructivism (mathematics)">constructivists</a> saw a need to improve upon the logical rigor in the <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">foundations of mathematics</a>.<a href="#cite_note-46">[f]</a> In the 1860s, <a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a> suggested a <a href="/wiki/Recursive_definition" title="Recursive definition">recursive definition</a> for natural numbers, thus stating they were not really natural—but a consequence of definitions. Later, two classes of such formal definitions emerged, using set theory and Peano's axioms respectively. Later still, they were shown to be equivalent in most practical applications. <a href="/wiki/Set-theoretical_definitions_of_natural_numbers" class="mw-redirect" title="Set-theoretical definitions of natural numbers">Set-theoretical definitions of natural numbers</a> were initiated by <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Frege</a>. He initially defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set. However, this definition turned out to lead to paradoxes, including <a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a>. To avoid such paradoxes, the formalism was modified so that a natural number is defined as a particular set, and any set that can be put into one-to-one correspondence with that set is said to have that number of elements.<a href="#cite_note-47">[41]</a> In 1881, <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a> provided the first <a href="/wiki/Axiomatic_system#Axiomatization" title="Axiomatic system">axiomatization</a> of natural-number arithmetic.<a href="#cite_note-48">[42]</a><a href="#cite_note-49">[43]</a> In 1888, <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> proposed another axiomatization of natural-number arithmetic,<a href="#cite_note-50">[44]</a> and in 1889, Peano published a simplified version of Dedekind's axioms in his book The principles of arithmetic presented by a new method (<a href="/wiki/Latin_language" class="mw-redirect" title="Latin language">Latin</a>: <a href="/wiki/Arithmetices_principia,_nova_methodo_exposita" title="Arithmetices principia, nova methodo exposita">Arithmetices principia, nova methodo exposita</a>). This approach is now called <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>. It is based on an <a href="/wiki/Axiomatization" class="mw-redirect" title="Axiomatization">axiomatization</a> of the properties of <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal numbers</a>: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is <a href="/wiki/Equiconsistent" class="mw-redirect" title="Equiconsistent">equiconsistent</a> with several weak systems of <a href="/wiki/Set_theory" title="Set theory">set theory</a>. One such system is <a href="/wiki/ZFC" class="mw-redirect" title="ZFC">ZFC</a> with the <a href="/wiki/Axiom_of_infinity" title="Axiom of infinity">axiom of infinity</a> replaced by its negation.<a href="#cite_note-51">[45]</a> Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include <a href="/wiki/Goodstein%27s_theorem" title="Goodstein's theorem">Goodstein's theorem</a>.<a href="#cite_note-52">[46]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><h2 id="Notation">Notation</h2> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=5" title="Edit section: Notation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> The <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of all natural numbers is standardly denoted N or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} .}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682f44bd6a1ea39ecf1e21a8290b9d5b2f504505" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} .}"></noscript><span class="lazy-image-placeholder" style="width: 2.325ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682f44bd6a1ea39ecf1e21a8290b9d5b2f504505" data-alt="{\displaystyle \mathbb {N} .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> <a href="#cite_note-:1-3">[2]</a><a href="#cite_note-53">[47]</a> Older texts have occasionally employed J as the symbol for this set.<a href="#cite_note-54">[48]</a> Since natural numbers may contain 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:<a href="#cite_note-ISO80000-41">[37]</a><a href="#cite_note-Grimaldi-55">[49]</a> <ul><li>Naturals without zero: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,...\}=\mathbb {N} ^{*}=\mathbb {N} ^{+}=\mathbb {N} _{0}\smallsetminus \{0\}=\mathbb {N} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,...\}=\mathbb {N} ^{*}=\mathbb {N} ^{+}=\mathbb {N} _{0}\smallsetminus \{0\}=\mathbb {N} _{1}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eb73372fb49aae41b93f74376dc418c0e749483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.927ex; height:3.009ex;" alt="{\displaystyle \{1,2,...\}=\mathbb {N} ^{*}=\mathbb {N} ^{+}=\mathbb {N} _{0}\smallsetminus \{0\}=\mathbb {N} _{1}}"></noscript><span class="lazy-image-placeholder" style="width: 39.927ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eb73372fb49aae41b93f74376dc418c0e749483" data-alt="{\displaystyle \{1,2,...\}=\mathbb {N} ^{*}=\mathbb {N} ^{+}=\mathbb {N} _{0}\smallsetminus \{0\}=\mathbb {N} _{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </li> <li>Naturals with zero: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\{0,1,2,...\}=\mathbb {N} _{0}=\mathbb {N} ^{0}=\mathbb {N} ^{*}\cup \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace"></mspace> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\{0,1,2,...\}=\mathbb {N} _{0}=\mathbb {N} ^{0}=\mathbb {N} ^{*}\cup \{0\}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ce4ab27eef4b83e333756d7b0a69c4ff3b4b57c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.223ex; height:3.176ex;" alt="{\displaystyle \;\{0,1,2,...\}=\mathbb {N} _{0}=\mathbb {N} ^{0}=\mathbb {N} ^{*}\cup \{0\}}"></noscript><span class="lazy-image-placeholder" style="width: 36.223ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ce4ab27eef4b83e333756d7b0a69c4ff3b4b57c" data-alt="{\displaystyle \;\{0,1,2,...\}=\mathbb {N} _{0}=\mathbb {N} ^{0}=\mathbb {N} ^{*}\cup \{0\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </li></ul> Alternatively, since the natural numbers naturally form a <a href="/wiki/Subset" title="Subset">subset</a> of the <a href="/wiki/Integer" title="Integer">integers</a> (often denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ), they may be referred to as the positive, or the non-negative integers, respectively.<a href="#cite_note-56">[50]</a> To be unambiguous about whether 0 is included or not, sometimes a superscript "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}"></noscript><span class="lazy-image-placeholder" style="width: 1.162ex;height: 1.509ex;vertical-align: 0.079ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" data-alt="{\displaystyle *}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> " or "+" is added in the former case, and a subscript (or superscript) "0" is added in the latter case:<a href="#cite_note-ISO80000-41">[37]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3,\dots \}=\{x\in \mathbb {Z} :x>0\}=\mathbb {Z} ^{+}=\mathbb {Z} _{>0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>:</mo> <mi>x</mi> <mo>></mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>></mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3,\dots \}=\{x\in \mathbb {Z} :x>0\}=\mathbb {Z} ^{+}=\mathbb {Z} _{>0}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da2e839bbf7dd82a83e4bf60c01b5a38c37e4096" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.45ex; height:3.009ex;" alt="{\displaystyle \{1,2,3,\dots \}=\{x\in \mathbb {Z} :x>0\}=\mathbb {Z} ^{+}=\mathbb {Z} _{>0}}"></noscript><span class="lazy-image-placeholder" style="width: 43.45ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da2e839bbf7dd82a83e4bf60c01b5a38c37e4096" data-alt="{\displaystyle \{1,2,3,\dots \}=\{x\in \mathbb {Z} :x>0\}=\mathbb {Z} ^{+}=\mathbb {Z} _{>0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1,2,\dots \}=\{x\in \mathbb {Z} :x\geq 0\}=\mathbb {Z} _{0}^{+}=\mathbb {Z} _{\geq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>:</mo> <mi>x</mi> <mo>≥</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,1,2,\dots \}=\{x\in \mathbb {Z} :x\geq 0\}=\mathbb {Z} _{0}^{+}=\mathbb {Z} _{\geq 0}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b34547645a96c1b4073616304df575d13c958523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.45ex; height:3.176ex;" alt="{\displaystyle \{0,1,2,\dots \}=\{x\in \mathbb {Z} :x\geq 0\}=\mathbb {Z} _{0}^{+}=\mathbb {Z} _{\geq 0}}"></noscript><span class="lazy-image-placeholder" style="width: 43.45ex;height: 3.176ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b34547645a96c1b4073616304df575d13c958523" data-alt="{\displaystyle \{0,1,2,\dots \}=\{x\in \mathbb {Z} :x\geq 0\}=\mathbb {Z} _{0}^{+}=\mathbb {Z} _{\geq 0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><h2 id="Properties">Properties</h2> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=6" title="Edit section: Properties" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> This section uses the convention <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} =\mathbb {N} _{0}=\mathbb {N} ^{*}\cup \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} =\mathbb {N} _{0}=\mathbb {N} ^{*}\cup \{0\}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddbef60796e9c37a573367bb6bcf04715861db92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.41ex; height:2.843ex;" alt="{\displaystyle \mathbb {N} =\mathbb {N} _{0}=\mathbb {N} ^{*}\cup \{0\}}"></noscript><span class="lazy-image-placeholder" style="width: 19.41ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddbef60796e9c37a573367bb6bcf04715861db92" data-alt="{\displaystyle \mathbb {N} =\mathbb {N} _{0}=\mathbb {N} ^{*}\cup \{0\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . <div class="mw-heading mw-heading3"><h3 id="Addition">Addition</h3> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=7" title="Edit section: Addition" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> Given the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></noscript><span class="lazy-image-placeholder" style="width: 1.678ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" data-alt="{\displaystyle \mathbb {N} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  of natural numbers and the <a href="/wiki/Successor_function" title="Successor function">successor function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\colon \mathbb {N} \to \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\colon \mathbb {N} \to \mathbb {N} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d8d6e80a52918c3b75f7062fe383b1e09b5e2c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.503ex; height:2.176ex;" alt="{\displaystyle S\colon \mathbb {N} \to \mathbb {N} }"></noscript><span class="lazy-image-placeholder" style="width: 9.503ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d8d6e80a52918c3b75f7062fe383b1e09b5e2c8" data-alt="{\displaystyle S\colon \mathbb {N} \to \mathbb {N} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  sending each natural number to the next one, one can define <a href="/wiki/Addition_in_N" class="mw-redirect" title="Addition in N">addition</a> of natural numbers recursively by setting a + 0 = a and a + S(b) = S(a + b) for all a, b. Thus, a + 1 = a + S(0) = S(a+0) = S(a), a + 2 = a + S(1) = S(a+1) = S(S(a)), and so on. The <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {N} ,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {N} ,+)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0072a6ee0ab943ce24dc44083bd60d50739a0b1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.329ex; height:2.843ex;" alt="{\displaystyle (\mathbb {N} ,+)}"></noscript><span class="lazy-image-placeholder" style="width: 6.329ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0072a6ee0ab943ce24dc44083bd60d50739a0b1f" data-alt="{\displaystyle (\mathbb {N} ,+)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is a <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a> <a href="/wiki/Monoid" title="Monoid">monoid</a> with <a href="/wiki/Identity_element" title="Identity element">identity element</a> 0. It is a <a href="/wiki/Free_object" title="Free object">free monoid</a> on one generator. This commutative monoid satisfies the <a href="/wiki/Cancellation_property" title="Cancellation property">cancellation property</a>, so it can be embedded in a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>. The smallest group containing the natural numbers is the <a href="/wiki/Integer" title="Integer">integers</a>. If 1 is defined as S(0), then b + 1 = b + S(0) = S(b + 0) = S(b). That is, b + 1 is simply the successor of b. <div class="mw-heading mw-heading3"><h3 id="Multiplication">Multiplication</h3> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=8" title="Edit section: Multiplication" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> Analogously, given that addition has been defined, a <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"></noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 1.509ex;vertical-align: 0.019ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" data-alt="{\displaystyle \times }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  can be defined via a × 0 = 0 and a × S(b) = (a × b) + a. This turns <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {N} ^{*},\times )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>,</mo> <mo>×</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {N} ^{*},\times )}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd9f6d877853c0e1b4e6477b70cac9644a5fe14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.384ex; height:2.843ex;" alt="{\displaystyle (\mathbb {N} ^{*},\times )}"></noscript><span class="lazy-image-placeholder" style="width: 7.384ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd9f6d877853c0e1b4e6477b70cac9644a5fe14" data-alt="{\displaystyle (\mathbb {N} ^{*},\times )}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  into a <a href="/wiki/Free_commutative_monoid" class="mw-redirect" title="Free commutative monoid">free commutative monoid</a> with identity element 1; a generator set for this monoid is the set of <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>. <div class="mw-heading mw-heading3"><h3 id="Relationship_between_addition_and_multiplication">Relationship between addition and multiplication</h3> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=9" title="Edit section: Relationship between addition and multiplication" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> Addition and multiplication are compatible, which is expressed in the <a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">distribution law</a>: a × (b + c) = (a × b) + (a × c). These properties of addition and multiplication make the natural numbers an instance of a <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a> <a href="/wiki/Semiring" title="Semiring">semiring</a>. 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></noscript><span class="lazy-image-placeholder" style="width: 1.678ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" data-alt="{\displaystyle \mathbb {N} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is not <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closed</a> under subtraction (that is, subtracting one natural from another does not always result in another natural), means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></noscript><span class="lazy-image-placeholder" style="width: 1.678ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" data-alt="{\displaystyle \mathbb {N} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is not a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>; instead it is a <a href="/wiki/Semiring" title="Semiring">semiring</a> (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 a + 1 = S(a) and a × 1 = a. Furthermore, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {N^{*}} ,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="double-struck">N</mi> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="double-struck">∗</mo> </mrow> </msup> </mrow> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {N^{*}} ,+)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22ad68f9dd3f2e7653a596f2e8d647a5de6b0bda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.384ex; height:2.843ex;" alt="{\displaystyle (\mathbb {N^{*}} ,+)}"></noscript><span class="lazy-image-placeholder" style="width: 7.384ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22ad68f9dd3f2e7653a596f2e8d647a5de6b0bda" data-alt="{\displaystyle (\mathbb {N^{*}} ,+)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  has no identity element. <div class="mw-heading mw-heading3"><h3 id="Order">Order</h3> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=10" title="Edit section: Order" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> In this section, juxtaposed variables such as ab indicate the product a × b,<a href="#cite_note-57">[51]</a> and the standard <a href="/wiki/Order_of_operations" title="Order of operations">order of operations</a> is assumed. A <a href="/wiki/Total_order" title="Total order">total order</a> on the natural numbers is defined by letting a ≤ b if and only if there exists another natural number c where a + c = b. This order is compatible with the <a href="/wiki/Arithmetical_operations" class="mw-redirect" title="Arithmetical operations">arithmetical operations</a> in the following sense: if a, b and c are natural numbers and a ≤ b, then a + c ≤ b + c and ac ≤ bc. An important property of the natural numbers is that they are <a href="/wiki/Well-order" title="Well-order">well-ordered</a>: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal number</a>; for the natural numbers, this is denoted as <a href="/wiki/Omega_(ordinal)" class="mw-redirect" title="Omega (ordinal)">ω</a> (omega). <div class="mw-heading mw-heading3"><h3 id="Division">Division</h3> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=11" title="Edit section: Division" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> In this section, juxtaposed variables such as ab indicate the product a × b, and the standard <a href="/wiki/Order_of_operations" title="Order of operations">order of operations</a> 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 <a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean division</a> is available as a substitute: for any two natural numbers a and b with b ≠ 0 there are natural numbers q and r such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=bq+r{\text{ and }}r<b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mi>q</mi> <mo>+</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>r</mi> <mo><</mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=bq+r{\text{ and }}r<b.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb374cf44fed4f1fad58fbf031da726405c28b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.985ex; height:2.509ex;" alt="{\displaystyle a=bq+r{\text{ and }}r<b.}"></noscript><span class="lazy-image-placeholder" style="width: 20.985ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb374cf44fed4f1fad58fbf031da726405c28b6" data-alt="{\displaystyle a=bq+r{\text{ and }}r<b.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> The number q is called the <a href="/wiki/Quotient" title="Quotient">quotient</a> and r is called the <a href="/wiki/Remainder" title="Remainder">remainder</a> of the division of a by b. The numbers q and r are uniquely determined by a and b. This Euclidean division is key to the several other properties (<a href="/wiki/Divisibility" class="mw-redirect" title="Divisibility">divisibility</a>), algorithms (such as the <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a>), and ideas in number theory. <div class="mw-heading mw-heading3"><h3 id="Algebraic_properties_satisfied_by_the_natural_numbers">Algebraic properties satisfied by the natural numbers</h3> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=12" title="Edit section: Algebraic properties satisfied by the natural numbers" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties: <ul><li><a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">Closure</a> under addition and multiplication: for all natural numbers a and b, both a + b and a × b are natural numbers.<a href="#cite_note-58">[52]</a></li> <li><a href="/wiki/Associativity" class="mw-redirect" title="Associativity">Associativity</a>: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.<a href="#cite_note-59">[53]</a></li> <li><a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">Commutativity</a>: for all natural numbers a and b, a + b = b + a and a × b = b × a.<a href="#cite_note-60">[54]</a></li> <li>Existence of <a href="/wiki/Identity_element" title="Identity element">identity elements</a>: for every natural number a, a + 0 = a and a × 1 = a. <ul><li>If the natural numbers are taken as "excluding 0", and "starting at 1", then for every natural number a, a × 1 = a. However, the "existence of additive identity element" property is not satisfied</li></ul></li> <li><a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">Distributivity</a> of multiplication over addition for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c).</li> <li>No nonzero <a href="/wiki/Zero_divisor" title="Zero divisor">zero divisors</a>: if a and b are natural numbers such that a × b = 0, then a = 0 or b = 0 (or both).</li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><h2 id="Generalizations">Generalizations</h2> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=13" title="Edit section: Generalizations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> Two important generalizations of natural numbers arise from the two uses of counting and ordering: <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal numbers</a> and <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal numbers</a>. <ul><li>A natural number can be used to express the size of a finite set; more precisely, a cardinal number is a measure for the size of a set, which is even suitable for infinite sets. The numbering of cardinals usually begins at zero, to accommodate the <a href="/wiki/Empty_set" title="Empty set">empty set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \emptyset }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }"></noscript><span class="lazy-image-placeholder" style="width: 1.162ex;height: 2.509ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" data-alt="{\displaystyle \emptyset }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . This concept of "size" relies on maps between sets, such that two sets have <a href="/wiki/Equinumerosity" title="Equinumerosity">the same size</a>, exactly if there exists a <a href="/wiki/Bijection" title="Bijection">bijection</a> between them. The set of natural numbers itself, and any bijective image of it, is said to be <a href="/wiki/Countable_set" title="Countable set">countably infinite</a> and to have <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> <a href="/wiki/Aleph_number#Aleph-null" title="Aleph number">aleph-null</a> (ℵ0).</li> <li>Natural numbers are also used as <a href="/wiki/Ordinal_numbers_(linguistics)" class="mw-redirect" title="Ordinal numbers (linguistics)">linguistic ordinal numbers</a>: "first", "second", "third", and so forth. The numbering of ordinals usually begins at zero, to accommodate the order type of the <a href="/wiki/Empty_set" title="Empty set">empty set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \emptyset }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }"></noscript><span class="lazy-image-placeholder" style="width: 1.162ex;height: 2.509ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" data-alt="{\displaystyle \emptyset }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any <a href="/wiki/Well-order" title="Well-order">well-ordered</a> countably infinite set without <a href="/wiki/Limit_points" class="mw-redirect" title="Limit points">limit points</a>. This assignment can be generalized to general well-orderings with a cardinality beyond countability, to yield the ordinal numbers. An ordinal number may also be used to describe the notion of "size" for a well-ordered set, in a sense different from cardinality: if there is an <a href="/wiki/Order_isomorphism" title="Order isomorphism">order isomorphism</a> (more than a bijection) between two well-ordered sets, they have the same ordinal number. The first ordinal number that is not a natural number is expressed as ω; this is also the ordinal number of the set of natural numbers itself.</li></ul> The least ordinal of cardinality ℵ0 (that is, the <a href="/wiki/Von_Neumann_cardinal_assignment" title="Von Neumann cardinal assignment">initial ordinal</a> of ℵ0) is ω but many well-ordered sets with cardinal number ℵ0 have an ordinal number greater than ω. For <a href="/wiki/Finite_set" title="Finite set">finite</a> well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, <a href="/wiki/Sequence" title="Sequence">sequence</a>. A countable <a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">non-standard model of arithmetic</a> satisfying the Peano Arithmetic (that is, the first-order Peano axioms) was developed by <a href="/wiki/Skolem" class="mw-redirect" title="Skolem">Skolem</a> in 1933. The <a href="/wiki/Hypernatural" class="mw-redirect" title="Hypernatural">hypernatural</a> numbers are an uncountable model that can be constructed from the ordinary natural numbers via the <a href="/wiki/Ultrapower_construction" class="mw-redirect" title="Ultrapower construction">ultrapower construction</a>. Other generalizations are discussed in <a href="/wiki/Number#Extensions_of_the_concept" title="Number">Number § Extensions of the concept</a>. <a href="/wiki/Georges_Reeb" title="Georges Reeb">Georges Reeb</a> used to claim provocatively that "The naïve integers don't fill up <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></noscript><span class="lazy-image-placeholder" style="width: 1.678ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" data-alt="{\displaystyle \mathbb {N} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ".<a href="#cite_note-61">[55]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><h2 id="Formal_definitions">Formal definitions</h2> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=14" title="Edit section: Formal definitions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> There are two standard methods for formally defining natural numbers. The first one, named for <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a>, consists of an autonomous <a href="/wiki/Axiomatic_theory" class="mw-redirect" title="Axiomatic theory">axiomatic theory</a> called <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>, based on few axioms called <a href="/wiki/Peano_axioms" title="Peano axioms">Peano axioms</a>. The second definition is based on <a href="/wiki/Set_theory" title="Set theory">set theory</a>. It defines the natural numbers as specific <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a>. More precisely, each natural number n is defined as an explicitly defined set, whose elements allow counting the elements of other sets, in the sense that the sentence "a set S has n elements" means that there exists a <a href="/wiki/One_to_one_correspondence" class="mw-redirect" title="One to one correspondence">one to one correspondence</a> between the two sets n and S. The sets used to define natural numbers satisfy Peano axioms. It follows that every <a href="/wiki/Theorem" title="Theorem">theorem</a> that can be stated and proved in Peano arithmetic can also be proved in set theory. However, the two definitions are not equivalent, as there are theorems that can be stated in terms of Peano arithmetic and proved in set theory, which are not provable inside Peano arithmetic. A probable example is <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a>. The definition of the integers as sets satisfying Peano axioms provide a <a href="/wiki/Model_(mathematical_logic)" class="mw-redirect" title="Model (mathematical logic)">model</a> of Peano arithmetic inside set theory. An important consequence is that, if set theory is <a href="/wiki/Consistent" class="mw-redirect" title="Consistent">consistent</a> (as it is usually guessed), then Peano arithmetic is consistent. In other words, if a contradiction could be proved in Peano arithmetic, then set theory would be contradictory, and every theorem of set theory would be both true and wrong. <div class="mw-heading mw-heading3"><h3 id="Peano_axioms">Peano axioms</h3> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=15" title="Edit section: Peano axioms" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Peano_axioms" title="Peano axioms">Peano axioms</a></div> The five Peano axioms are the following:<a href="#cite_note-62">[56]</a><a href="#cite_note-63">[g]</a> <ol><li>0 is a natural number.</li> <li>Every natural number has a successor which is also a natural number.</li> <li>0 is not the successor of any natural number.</li> <li>If the successor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  equals the successor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></noscript><span class="lazy-image-placeholder" style="width: 1.155ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" data-alt="{\displaystyle y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></noscript><span class="lazy-image-placeholder" style="width: 1.155ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" data-alt="{\displaystyle y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</li> <li>The <a href="/wiki/Axiom_of_induction" class="mw-redirect" title="Axiom of induction">axiom of induction</a>: If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.</li></ol> These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic, the successor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+1}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16df430ed7a23df9b160a5bbd957f306a0c3baa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.333ex; height:2.343ex;" alt="{\displaystyle x+1}"></noscript><span class="lazy-image-placeholder" style="width: 5.333ex;height: 2.343ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16df430ed7a23df9b160a5bbd957f306a0c3baa7" data-alt="{\displaystyle x+1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . <div class="mw-heading mw-heading3"><h3 id="Set-theoretic_definition">Set-theoretic definition</h3> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=16" title="Edit section: Set-theoretic definition" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Set-theoretic_definition_of_natural_numbers" title="Set-theoretic definition of natural numbers">Set-theoretic definition of natural numbers</a></div> Intuitively, the natural number n is the common property of all <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> that have n elements. So, it seems natural to define n as an <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence class</a> under the relation "can be made in <a href="/wiki/One_to_one_correspondence" class="mw-redirect" title="One to one correspondence">one to one correspondence</a>". This does not work in all <a href="/wiki/Set_theory" title="Set theory">set theories</a>, as such an equivalence class would not be a set<a href="#cite_note-64">[h]</a> (because of <a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a>). The standard solution is to define a particular set with n elements that will be called the natural number n. The following definition was first published by <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a>,<a href="#cite_note-vonNeumann1923pp199-208-65">[57]</a> although Levy attributes the idea to unpublished work of Zermelo in 1916.<a href="#cite_note-Levy-66">[58]</a> As this definition extends to <a href="/wiki/Infinite_set" title="Infinite set">infinite set</a> as a definition of <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal number</a>, the sets considered below are sometimes called <a href="/wiki/Von_Neumann_ordinals" class="mw-redirect" title="Von Neumann ordinals">von Neumann ordinals</a>. The definition proceeds as follows: <ul><li>Call 0 = { }, the <a href="/wiki/Empty_set" title="Empty set">empty set</a>.</li> <li>Define the successor S(a) of any set a by S(a) = a ∪ {a}.</li> <li>By the <a href="/wiki/Axiom_of_infinity" title="Axiom of infinity">axiom of infinity</a>, there exist sets which contain 0 and are <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closed</a> under the successor function. Such sets are said to be inductive. The intersection of all inductive sets is still an inductive set.</li> <li>This intersection is the set of the natural numbers.</li></ul> It follows that the natural numbers are defined iteratively as follows: <dl><dd><ul><li>0 = { },</li> <li>1 = 0 ∪ {0} = {0} = {{ }},</li> <li>2 = 1 ∪ {1} = {0, 1} = {{ }, {{ }}},</li> <li>3 = 2 ∪ {2} = {0, 1, 2} = {{ }, {{ }}, {{ }, {{ }}}},</li> <li>n = n−1 ∪ {n−1} = {0, 1, ..., n−1} = {{ }, {{ }}, ..., {{ }, {{ }}, ...}},</li> <li>etc.</li></ul></dd></dl> It can be checked that the natural numbers satisfy the <a href="/wiki/Peano_axioms" title="Peano axioms">Peano axioms</a>. With this definition, given a natural number n, the sentence "a set S has n elements" can be formally defined as "there exists a <a href="/wiki/Bijection" title="Bijection">bijection</a> from n to S." This formalizes the operation of counting the elements of S. Also, n ≤ m if and only if n is a <a href="/wiki/Subset" title="Subset">subset</a> of m. In other words, the <a href="/wiki/Set_inclusion" class="mw-redirect" title="Set inclusion">set inclusion</a> defines the usual <a href="/wiki/Total_order" title="Total order">total order</a> on the natural numbers. This order is a <a href="/wiki/Well-order" title="Well-order">well-order</a>. It follows from the definition that each natural number is equal to the set of all natural numbers less than it. This definition, can be extended to the <a href="/wiki/Von_Neumann_ordinal" class="mw-redirect" title="Von Neumann ordinal">von Neumann definition of ordinals</a> for defining all <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal numbers</a>, including the infinite ones: "each ordinal is the well-ordered set of all smaller ordinals." If one <a href="/wiki/Finitism" title="Finitism">does not accept the axiom of infinity</a>, the natural numbers may not form a set. Nevertheless, the natural numbers can still be individually defined as above, and they still satisfy the Peano axioms. There are other set theoretical constructions. In particular, <a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Ernst Zermelo</a> provided a construction that is nowadays only of historical interest, and is sometimes referred to as <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style>Zermelo ordinals.<a href="#cite_note-Levy-66">[58]</a> It consists in defining 0 as the empty set, and S(a) = {a}. With this definition each nonzero natural number is a <a href="/wiki/Singleton_set" class="mw-redirect" title="Singleton set">singleton set</a>. So, the property of the natural numbers to represent <a href="/wiki/Cardinalities" class="mw-redirect" title="Cardinalities">cardinalities</a> is not directly accessible; only the ordinal property (being the nth element of a sequence) is immediate. Unlike von Neumann's construction, the Zermelo ordinals do not extend to infinite ordinals. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><h2 id="See_also">See also</h2> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=17" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><noscript><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" data-file-width="128" data-file-height="128"></noscript><span class="lazy-image-placeholder" style="width: 28px;height: 28px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" data-alt="icon" data-width="28" data-height="28" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-class="mw-file-element"> </a><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul> <ul><li><a href="/wiki/Canonical_representation_of_a_positive_integer" class="mw-redirect" title="Canonical representation of a positive integer">Canonical representation of a positive integer</a> – Representation of a number as a product of primes</li> <li><a href="/wiki/Countable_set" title="Countable set">Countable set</a> – Mathematical set that can be enumerated</li> <li><a href="/wiki/Sequence" title="Sequence">Sequence</a> – Function of the natural numbers in another set</li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a> – Generalization of "n-th" to infinite cases</li> <li><a href="/wiki/Cardinal_number" title="Cardinal number">Cardinal number</a> – Size of a possibly infinite set</li> <li><a href="/wiki/Set-theoretic_definition_of_natural_numbers" title="Set-theoretic definition of natural numbers">Set-theoretic definition of natural numbers</a> – Axiom(s) of Set Theory</li></ul> <table style="margin:2em; border:2px solid silver; font-size:95%; border-collapse:collapse"> <tbody><tr> <td> <table style="margin:4px; border:2px solid silver"> <tbody><tr> <td> <table style="margin:1em"> <caption><a href="/wiki/Number_system" class="mw-redirect" title="Number system">Number systems</a> </caption> <tbody><tr> <td><a href="/wiki/Complex_number" title="Complex number">Complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {C} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c800b917bd652c093461395df2d796718aef00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {C} }"></noscript><span class="lazy-image-placeholder" style="width: 3.615ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c800b917bd652c093461395df2d796718aef00" data-alt="{\displaystyle :\;\mathbb {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a href="/wiki/Real_number" title="Real number">Real</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {R} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b09bba427588b2a529ebcf8fdb7536da42003b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {R} }"></noscript><span class="lazy-image-placeholder" style="width: 3.615ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b09bba427588b2a529ebcf8fdb7536da42003b1" data-alt="{\displaystyle :\;\mathbb {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a href="/wiki/Rational_number" title="Rational number">Rational</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {Q} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f77b368ade52a03084dad12fba5b25129cebe0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.745ex; height:2.509ex;" alt="{\displaystyle :\;\mathbb {Q} }"></noscript><span class="lazy-image-placeholder" style="width: 3.745ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f77b368ade52a03084dad12fba5b25129cebe0d" data-alt="{\displaystyle :\;\mathbb {Q} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a href="/wiki/Integer" title="Integer">Integer</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cff631a0751189f28ca66b5d8ab161f05259f8f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 3.487ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cff631a0751189f28ca66b5d8ab161f05259f8f1" data-alt="{\displaystyle :\;\mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td> <table> <tbody><tr> <td><a class="mw-selflink selflink">Natural</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :\;\mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :\;\mathbb {N} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51ba123110cb54a0b89909e10845ed2ee8c52e8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:2.176ex;" alt="{\displaystyle :\;\mathbb {N} }"></noscript><span class="lazy-image-placeholder" style="width: 3.615ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51ba123110cb54a0b89909e10845ed2ee8c52e8f" data-alt="{\displaystyle :\;\mathbb {N} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td><a href="/wiki/Zero" class="mw-redirect" title="Zero">Zero</a>: 0 </td></tr> <tr> <td><a href="/wiki/One" class="mw-redirect" title="One">One</a>: 1 </td></tr> <tr> <td><a href="/wiki/Prime_number" title="Prime number">Prime numbers</a> </td></tr> <tr> <td><a href="/wiki/Composite_number" title="Composite number">Composite numbers</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td><a href="/wiki/Negative_integer" class="mw-redirect" title="Negative integer">Negative integers</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td> <table> <tbody><tr> <td><a href="/wiki/Fraction" title="Fraction">Fraction</a> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td><a href="/wiki/Finite_decimal" class="mw-redirect" title="Finite decimal">Finite decimal</a> </td></tr> <tr> <td><a href="/wiki/Dyadic_rational" title="Dyadic rational">Dyadic (finite binary)</a> </td></tr> <tr> <td><a href="/wiki/Repeating_decimal" title="Repeating decimal">Repeating decimal</a> </td> <td> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td> <table> <tbody><tr> <td><a href="/wiki/Irrational_number" title="Irrational number">Irrational</a> </td> <td> <table style="border-left:4px solid green"> <tbody><tr> <td><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic irrational</a> </td></tr> <tr> <td><a href="/wiki/Period_(algebraic_geometry)" title="Period (algebraic geometry)">Irrational period</a> </td></tr> <tr> <td><a href="/wiki/Transcendental_number" title="Transcendental number">Transcendental</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr> <tr> <td><a href="/wiki/Imaginary_number" title="Imaginary number">Imaginary</a> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><h2 id="Notes">Notes</h2> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=18" title="Edit section: Notes" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-7 collapsible-block" id="mf-section-7"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-2"><a href="#cite_ref-2">^</a> See <a href="#Emergence_as_a_term">§ Emergence as a term</a> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> A tablet found at Kish ... thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place.<a href="#cite_note-13">[12]</a> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> This convention is used, for example, in <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a>, see D. Joyce's web edition of Book VII.<a href="#cite_note-EuclidVIIJoyce-18">[16]</a> </li> <li id="cite_note-MacLaneBirkhoff1999p15-32"><a href="#cite_ref-MacLaneBirkhoff1999p15_32-0">^</a> <a href="#CITEREFMac_LaneBirkhoff1999">Mac Lane & Birkhoff (1999</a>, p. 15) include zero in the natural numbers: 'Intuitively, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} =\{0,1,2,\ldots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} =\{0,1,2,\ldots \}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6259fa720c155df5af3c96fda39822267e8bf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.414ex; height:2.843ex;" alt="{\displaystyle \mathbb {N} =\{0,1,2,\ldots \}}"></noscript><span class="lazy-image-placeholder" style="width: 16.414ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6259fa720c155df5af3c96fda39822267e8bf0" data-alt="{\displaystyle \mathbb {N} =\{0,1,2,\ldots \}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  of all natural numbers may be described as follows: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></noscript><span class="lazy-image-placeholder" style="width: 1.678ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" data-alt="{\displaystyle \mathbb {N} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  contains an "initial" number 0; ...'. They follow that with their version of the <a href="/wiki/Peano%27s_axioms" class="mw-redirect" title="Peano's axioms">Peano's axioms</a>. </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> The English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891–1892, 19, quoting from a lecture of Kronecker's of 1886."<a href="#cite_note-43">[39]</a><a href="#cite_note-44">[40]</a> </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> "Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." (<a href="#CITEREFEves1990">Eves 1990</a>, p. 606) </li> <li id="cite_note-63"><a href="#cite_ref-63">^</a> <a href="#CITEREFHamilton1988">Hamilton (1988</a>, pp. 117 ff) calls them "Peano's Postulates" and begins with "1. 0 is a natural number." <a href="#CITEREFHalmos1960">Halmos (1960</a>, p. 46) uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I) 0 ∈ ω (where, of course, 0 = ∅" (ω is the set of all natural numbers). <a href="#CITEREFMorash1991">Morash (1991)</a> gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1: An Axiomatization for the System of Positive Integers) </li> <li id="cite_note-64"><a href="#cite_ref-64">^</a> In some set theories, e.g., <a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a>, a <a href="/wiki/Universal_set" title="Universal set">universal set</a> exists and Russel's paradox cannot be formulated. </li> </ol></div></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"><h2 id="References">References</h2> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=19" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-8 collapsible-block" id="mf-section-8"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-Enderton-1">^ <a href="#cite_ref-Enderton_1-0">a</a> <a href="#cite_ref-Enderton_1-1">b</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFEnderton1977" class="citation book cs1">Enderton, Herbert B. (1977). Elements of set theory. New York: Academic Press. p. 66. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0122384407" title="Special:BookSources/0122384407"><bdi>0122384407</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+set+theory&rft.place=New+York&rft.pages=66&rft.pub=Academic+Press&rft.date=1977&rft.isbn=0122384407&rft.aulast=Enderton&rft.aufirst=Herbert+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-:1-3">^ <a href="#cite_ref-:1_3-0">a</a> <a href="#cite_ref-:1_3-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/NaturalNumber.html">"Natural Number"</a>. mathworld.wolfram.com. Retrieved 11 August 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Natural+Number&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FNaturalNumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGanssleBarr2003" class="citation encyclopaedia cs1">Ganssle, Jack G. & Barr, Michael (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zePGx82d_fwC">"integer"</a>. Embedded Systems Dictionary. Taylor & Francis. pp. 138 (integer), 247 (signed integer), & 276 (unsigned integer). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-57820-120-4" title="Special:BookSources/978-1-57820-120-4"><bdi>978-1-57820-120-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170329150719/https://books.google.com/books?id=zePGx82d_fwC">Archived</a> from the original on 29 March 2017. Retrieved 28 March 2017 – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=integer&rft.btitle=Embedded+Systems+Dictionary&rft.pages=138+%28integer%29%2C+247+%28signed+integer%29%2C+%26+276+%28unsigned+integer%29&rft.pub=Taylor+%26+Francis&rft.date=2003&rft.isbn=978-1-57820-120-4&rft.aulast=Ganssle&rft.aufirst=Jack+G.&rft.au=Barr%2C+Michael&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzePGx82d_fwC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-MathWorld_CountingNumber-5"><a href="#cite_ref-MathWorld_CountingNumber_5-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/CountingNumber.html">"Counting Number"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Counting+Number&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCountingNumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWoodinWinter2024" class="citation journal cs1">Woodin, Greg; Winter, Bodo (2024). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11475258">"Numbers in Context: Cardinals, Ordinals, and Nominals in American English"</a>. Cognitive Science. 48 (6) e13471. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fcogs.13471">10.1111/cogs.13471</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11475258">11475258</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Cognitive+Science&rft.atitle=Numbers+in+Context%3A+Cardinals%2C+Ordinals%2C+and+Nominals+in+American+English&rft.volume=48&rft.issue=6&rft.artnum=e13471&rft.date=2024&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC11475258%23id-name%3DPMC&rft_id=info%3Adoi%2F10.1111%2Fcogs.13471&rft.aulast=Woodin&rft.aufirst=Greg&rft.au=Winter%2C+Bodo&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC11475258&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFMendelson2008">Mendelson (2008</a>, p. x) says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers." </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFBluman2010">Bluman (2010</a>, p. 1): "Numbers make up the foundation of mathematics." </li> <li id="cite_note-RBINS_intro-9"><a href="#cite_ref-RBINS_intro_9-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304051733/https://www.naturalsciences.be/expo/old_ishango/en/ishango/introduction.html">"Introduction"</a>. <a href="/wiki/Ishango_bone" title="Ishango bone">Ishango bone</a>. Brussels, Belgium: <a href="/wiki/Royal_Belgian_Institute_of_Natural_Sciences" class="mw-redirect" title="Royal Belgian Institute of Natural Sciences">Royal Belgian Institute of Natural Sciences</a>. Archived from <a rel="nofollow" class="external text" href="https://www.naturalsciences.be/expo/old_ishango/en/ishango/introduction.html">the original</a> on 4 March 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Introduction&rft.place=Brussels%2C+Belgium&rft.series=Ishango+bone&rft.pub=Royal+Belgian+Institute+of+Natural+Sciences&rft_id=https%3A%2F%2Fwww.naturalsciences.be%2Fexpo%2Fold_ishango%2Fen%2Fishango%2Fintroduction.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-RBINS_flash-10"><a href="#cite_ref-RBINS_flash_10-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160527164619/http://ishango.naturalsciences.be/Flash/flash_local/Ishango-02-EN.html">"Flash presentation"</a>. <a href="/wiki/Ishango_bone" title="Ishango bone">Ishango bone</a>. Brussels, Belgium: <a href="/wiki/Royal_Belgian_Institute_of_Natural_Sciences" class="mw-redirect" title="Royal Belgian Institute of Natural Sciences">Royal Belgian Institute of Natural Sciences</a>. Archived from <a rel="nofollow" class="external text" href="http://ishango.naturalsciences.be/Flash/flash_local/Ishango-02-EN.html">the original</a> on 27 May 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Flash+presentation&rft.place=Brussels%2C+Belgium&rft.series=Ishango+bone&rft.pub=Royal+Belgian+Institute+of+Natural+Sciences&rft_id=http%3A%2F%2Fishango.naturalsciences.be%2FFlash%2Fflash_local%2FIshango-02-EN.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-UNESCO-11"><a href="#cite_ref-UNESCO_11-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20141110195426/http://www2.astronomicalheritage.net/index.php/show-entity?identity=4&idsubentity=1">"The Ishango Bone, Democratic Republic of the Congo"</a>. <a href="/wiki/UNESCO" title="UNESCO">UNESCO</a>'s Portal to the Heritage of Astronomy. Archived from <a rel="nofollow" class="external text" href="http://www2.astronomicalheritage.net/index.php/show-entity?identity=4&idsubentity=1">the original</a> on 10 November 2014.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=UNESCO%27s+Portal+to+the+Heritage+of+Astronomy&rft.atitle=The+Ishango+Bone%2C+Democratic+Republic+of+the+Congo&rft_id=http%3A%2F%2Fwww2.astronomicalheritage.net%2Findex.php%2Fshow-entity%3Fidentity%3D4%26idsubentity%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988">, on permanent display at the <a href="/wiki/Royal_Belgian_Institute_of_Natural_Sciences" class="mw-redirect" title="Royal Belgian Institute of Natural Sciences">Royal Belgian Institute of Natural Sciences</a>, Brussels, Belgium. </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIfrah2000" class="citation book cs1">Ifrah, Georges (2000). The Universal History of Numbers. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-37568-3" title="Special:BookSources/0-471-37568-3"><bdi>0-471-37568-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Universal+History+of+Numbers&rft.pub=Wiley&rft.date=2000&rft.isbn=0-471-37568-3&rft.aulast=Ifrah&rft.aufirst=Georges&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html">"A history of Zero"</a>. MacTutor History of Mathematics. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130119083234/http://www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html">Archived</a> from the original on 19 January 2013. Retrieved 23 January 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MacTutor+History+of+Mathematics&rft.atitle=A+history+of+Zero&rft_id=http%3A%2F%2Fwww-history.mcs.st-and.ac.uk%2Fhistory%2FHistTopics%2FZero.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMann2005" class="citation book cs1">Mann, Charles C. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Jw2TE_UNHJYC&pg=PA19">1491: New Revelations of the Americas before Columbus</a>. Knopf. p. 19. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4000-4006-3" title="Special:BookSources/978-1-4000-4006-3"><bdi>978-1-4000-4006-3</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150514105855/https://books.google.com/books?id=Jw2TE_UNHJYC&pg=PA19">Archived</a> from the original on 14 May 2015. Retrieved 3 February 2015 – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=1491%3A+New+Revelations+of+the+Americas+before+Columbus&rft.pages=19&rft.pub=Knopf&rft.date=2005&rft.isbn=978-1-4000-4006-3&rft.aulast=Mann&rft.aufirst=Charles+C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJw2TE_UNHJYC%26pg%3DPA19&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEvans2014" class="citation book cs1">Evans, Brian (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3CPwAgAAQBAJ&pg=PT73">"Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations"</a>. The Development of Mathematics Throughout the Centuries: A brief history in a cultural context. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-85397-9" title="Special:BookSources/978-1-118-85397-9"><bdi>978-1-118-85397-9</bdi></a> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+10.+Pre-Columbian+Mathematics%3A+The+Olmec%2C+Maya%2C+and+Inca+Civilizations&rft.btitle=The+Development+of+Mathematics+Throughout+the+Centuries%3A+A+brief+history+in+a+cultural+context&rft.pub=John+Wiley+%26+Sons&rft.date=2014&rft.isbn=978-1-118-85397-9&rft.aulast=Evans&rft.aufirst=Brian&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3CPwAgAAQBAJ%26pg%3DPT73&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeckers2003" class="citation web cs1">Deckers, Michael (25 August 2003). <a rel="nofollow" class="external text" href="http://hbar.phys.msu.ru/gorm/chrono/paschata.htm">"Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius"</a>. Hbar.phys.msu.ru. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190115083618/http://hbar.phys.msu.ru/gorm/chrono/paschata.htm">Archived</a> from the original on 15 January 2019. Retrieved 13 February 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Cyclus+Decemnovennalis+Dionysii+%E2%80%93+Nineteen+year+cycle+of+Dionysius&rft.pub=Hbar.phys.msu.ru&rft.date=2003-08-25&rft.aulast=Deckers&rft.aufirst=Michael&rft_id=http%3A%2F%2Fhbar.phys.msu.ru%2Fgorm%2Fchrono%2Fpaschata.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-EuclidVIIJoyce-18"><a href="#cite_ref-EuclidVIIJoyce_18-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEuclid" class="citation book cs1"><a href="/wiki/Euclid" title="Euclid">Euclid</a>. <a rel="nofollow" class="external text" href="http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/defVII1.html">"Book VII, definitions 1 and 2"</a>. In Joyce, D. (ed.). <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a>. Clark University.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Book+VII%2C+definitions+1+and+2&rft.btitle=Elements&rft.pub=Clark+University&rft.au=Euclid&rft_id=http%3A%2F%2Faleph0.clarku.edu%2F~djoyce%2Fjava%2Felements%2FbookVII%2FdefVII1.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-Mueller_2006_p._58-20"><a href="#cite_ref-Mueller_2006_p._58_20-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMueller2006" class="citation book cs1">Mueller, Ian (2006). Philosophy of mathematics and deductive structure in <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a>. Mineola, New York: Dover Publications. p. 58. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-45300-2" title="Special:BookSources/978-0-486-45300-2"><bdi>978-0-486-45300-2</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/69792712">69792712</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Philosophy+of+mathematics+and+deductive+structure+in+Euclid%27s+Elements&rft.place=Mineola%2C+New+York&rft.pages=58&rft.pub=Dover+Publications&rft.date=2006&rft_id=info%3Aoclcnum%2F69792712&rft.isbn=978-0-486-45300-2&rft.aulast=Mueller&rft.aufirst=Ian&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEuclid" class="citation book cs1"><a href="/wiki/Euclid" title="Euclid">Euclid</a>. <a rel="nofollow" class="external text" href="http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/defVII22.html">"Book VII, definition 22"</a>. In Joyce, D. (ed.). <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a>. Clark University. <q>A perfect number is that which is equal to the sum of its own parts.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Book+VII%2C+definition+22&rft.btitle=Elements&rft.pub=Clark+University&rft.au=Euclid&rft_id=http%3A%2F%2Faleph0.clarku.edu%2F~djoyce%2Fjava%2Felements%2FbookVII%2FdefVII22.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> In definition VII.3 a "part" was defined as a number, but here 1 is considered to be a part, so that for example 6 = 1 + 2 + 3 is a perfect number. </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKline1990" class="citation book cs1">Kline, Morris (1990) [1972]. Mathematical Thought from Ancient to Modern Times. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-506135-7" title="Special:BookSources/0-19-506135-7"><bdi>0-19-506135-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Thought+from+Ancient+to+Modern+Times&rft.pub=Oxford+University+Press&rft.date=1990&rft.isbn=0-19-506135-7&rft.aulast=Kline&rft.aufirst=Morris&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChuquet1881" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Nicolas_Chuquet" title="Nicolas Chuquet">Chuquet, Nicolas</a> (1881) [1484]. <a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k62599266/f75.image">Le Triparty en la science des nombres</a> (in French).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Le+Triparty+en+la+science+des+nombres&rft.date=1881&rft.aulast=Chuquet&rft.aufirst=Nicolas&rft_id=https%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k62599266%2Ff75.image&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEmerson1763" class="citation book cs1">Emerson, William (1763). <a rel="nofollow" class="external text" href="https://archive.org/details/bim_eighteenth-century_the-method-of-increments_emerson-william_1763/page/112/mode/2up">The method of increments</a>. p. 113.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+method+of+increments&rft.pages=113&rft.date=1763&rft.aulast=Emerson&rft.aufirst=William&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbim_eighteenth-century_the-method-of-increments_emerson-william_1763%2Fpage%2F112%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-MacTutor-25">^ <a href="#cite_ref-MacTutor_25-0">a</a> <a href="#cite_ref-MacTutor_25-1">b</a> <a href="#cite_ref-MacTutor_25-2">c</a> <a href="#cite_ref-MacTutor_25-3">d</a> <a href="#cite_ref-MacTutor_25-4">e</a> <a href="#cite_ref-MacTutor_25-5">f</a> <a href="#cite_ref-MacTutor_25-6">g</a> <a href="#cite_ref-MacTutor_25-7">h</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Miller/mathword/n/">"Earliest Known Uses of Some of the Words of Mathematics (N)"</a>. Maths History.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Maths+History&rft.atitle=Earliest+Known+Uses+of+Some+of+the+Words+of+Mathematics+%28N%29&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FMiller%2Fmathword%2Fn%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFontenelle1727" class="citation book cs1 cs1-prop-foreign-lang-source">Fontenelle, Bernard de (1727). <a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k64762n/f31.item">Eléments de la géométrie de l'infini</a> (in French). p. 3.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=El%C3%A9ments+de+la+g%C3%A9om%C3%A9trie+de+l%27infini&rft.pages=3&rft.date=1727&rft.aulast=Fontenelle&rft.aufirst=Bernard+de&rft_id=https%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k64762n%2Ff31.item&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1 cs1-prop-foreign-lang-source"><a rel="nofollow" class="external text" href="https://archive.org/details/arithmeticespri00peangoog/page/n12/mode/2up">Arithmetices principia: nova methodo</a> (in Latin). Fratres Bocca. 1889. p. 12.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Arithmetices+principia%3A+nova+methodo&rft.pages=12&rft.pub=Fratres+Bocca&rft.date=1889&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Farithmeticespri00peangoog%2Fpage%2Fn12%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeano1901" class="citation book cs1 cs1-prop-foreign-lang-source">Peano, Giuseppe (1901). <a rel="nofollow" class="external text" href="https://archive.org/details/formulairedesmat00pean/page/38/mode/2up">Formulaire des mathematiques</a> (in French). Paris, Gauthier-Villars. p. 39.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Formulaire+des+mathematiques&rft.pages=39&rft.pub=Paris%2C+Gauthier-Villars&rft.date=1901&rft.aulast=Peano&rft.aufirst=Giuseppe&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fformulairedesmat00pean%2Fpage%2F38%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFine1904" class="citation book cs1">Fine, Henry Burchard (1904). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RR4PAAAAIAAJ&dq=%22natural%20number%22&pg=PA6">A College Algebra</a>. Ginn. p. 6.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+College+Algebra&rft.pages=6&rft.pub=Ginn&rft.date=1904&rft.aulast=Fine&rft.aufirst=Henry+Burchard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRR4PAAAAIAAJ%26dq%3D%2522natural%2520number%2522%26pg%3DPA6&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=184i06Py1ZYC&dq=%22natural%20number%22%201&pg=PA12">Advanced Algebra: A Study Guide to be Used with USAFI Course MC 166 Or CC166</a>. United States Armed Forces Institute. 1958. p. 12.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Algebra%3A+A+Study+Guide+to+be+Used+with+USAFI+Course+MC+166+Or+CC166&rft.pages=12&rft.pub=United+States+Armed+Forces+Institute&rft.date=1958&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D184i06Py1ZYC%26dq%3D%2522natural%2520number%2522%25201%26pg%3DPA12&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://archive.lib.msu.edu/crcmath/math/math/n/n035.htm">"Natural Number"</a>. archive.lib.msu.edu.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=archive.lib.msu.edu&rft.atitle=Natural+Number&rft_id=https%3A%2F%2Farchive.lib.msu.edu%2Fcrcmath%2Fmath%2Fmath%2Fn%2Fn035.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-Křížek-33">^ <a href="#cite_ref-K%C5%99%C3%AD%C5%BEek_33-0">a</a> <a href="#cite_ref-K%C5%99%C3%AD%C5%BEek_33-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKřížekSomerŠolcová2021" class="citation book cs1">Křížek, Michal; Somer, Lawrence; Šolcová, Alena (21 September 2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tklEEAAAQBAJ&dq=natural%20numbers%20zero&pg=PA6">From Great Discoveries in Number Theory to Applications</a>. Springer Nature. p. 6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-83899-7" title="Special:BookSources/978-3-030-83899-7"><bdi>978-3-030-83899-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Great+Discoveries+in+Number+Theory+to+Applications&rft.pages=6&rft.pub=Springer+Nature&rft.date=2021-09-21&rft.isbn=978-3-030-83899-7&rft.aulast=K%C5%99%C3%AD%C5%BEek&rft.aufirst=Michal&rft.au=Somer%2C+Lawrence&rft.au=%C5%A0olcov%C3%A1%2C+Alena&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtklEEAAAQBAJ%26dq%3Dnatural%2520numbers%2520zero%26pg%3DPA6&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> See, for example, <a href="#CITEREFCarothers2000">Carothers (2000</a>, p. 3) or <a href="#CITEREFThomsonBrucknerBruckner2008">Thomson, Bruckner & Bruckner (2008</a>, p. 2) </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGowers2008" class="citation book cs1">Gowers, Timothy (2008). The Princeton companion to mathematics. Princeton: Princeton university press. p. 17. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11880-2" title="Special:BookSources/978-0-691-11880-2"><bdi>978-0-691-11880-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Princeton+companion+to+mathematics&rft.place=Princeton&rft.pages=17&rft.pub=Princeton+university+press&rft.date=2008&rft.isbn=978-0-691-11880-2&rft.aulast=Gowers&rft.aufirst=Timothy&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBagaria2017" class="citation book cs1">Bagaria, Joan (2017). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/set-theory/">Set Theory</a> (Winter 2014 ed.). The Stanford Encyclopedia of Philosophy. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150314173026/http://plato.stanford.edu/entries/set-theory/">Archived</a> from the original on 14 March 2015. Retrieved 13 February 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory&rft.edition=Winter+2014&rft.pub=The+Stanford+Encyclopedia+of+Philosophy&rft.date=2017&rft.aulast=Bagaria&rft.aufirst=Joan&rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Fset-theory%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldrei1998" class="citation book cs1">Goldrei, Derek (1998). "3". <a rel="nofollow" class="external text" href="https://archive.org/details/classicsettheory00gold">Classic Set Theory: A guided independent study</a> (1. ed., 1. print ed.). Boca Raton, Fla. [u.a.]: Chapman & Hall/CRC. p. <a rel="nofollow" class="external text" href="https://archive.org/details/classicsettheory00gold/page/n39">33</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-412-60610-6" title="Special:BookSources/978-0-412-60610-6"><bdi>978-0-412-60610-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=3&rft.btitle=Classic+Set+Theory%3A+A+guided+independent+study&rft.place=Boca+Raton%2C+Fla.+%5Bu.a.%5D&rft.pages=33&rft.edition=1.+ed.%2C+1.+print&rft.pub=Chapman+%26+Hall%2FCRC&rft.date=1998&rft.isbn=978-0-412-60610-6&rft.aulast=Goldrei&rft.aufirst=Derek&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fclassicsettheory00gold&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="http://www.merriam-webster.com/dictionary/natural%20number">"natural number"</a>. Merriam-Webster.com. <a href="/wiki/Merriam-Webster" title="Merriam-Webster">Merriam-Webster</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191213133201/https://www.merriam-webster.com/dictionary/natural%20number">Archived</a> from the original on 13 December 2019. Retrieved 4 October 2014.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=natural+number&rft.btitle=Merriam-Webster.com&rft.pub=Merriam-Webster&rft_id=http%3A%2F%2Fwww.merriam-webster.com%2Fdictionary%2Fnatural%2520number&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrown1978" class="citation journal cs1">Brown, Jim (1978). "In defense of index origin 0". 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Retrieved 19 January 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=jsoftware.com&rft.atitle=Is+index+origin+0+a+hindrance%3F&rft.aulast=Hui&rft.aufirst=Roger&rft_id=http%3A%2F%2Fwww.jsoftware.com%2Fpapers%2Findexorigin.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-ISO80000-41">^ <a href="#cite_ref-ISO80000_41-0">a</a> <a href="#cite_ref-ISO80000_41-1">b</a> <a href="#cite_ref-ISO80000_41-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://cdn.standards.iteh.ai/samples/64973/329519100abd447ea0d49747258d1094/ISO-80000-2-2019.pdf#page=10">"Standard number sets and intervals"</a> (PDF). <a rel="nofollow" class="external text" href="https://www.iso.org/standard/64973.html">ISO 80000-2:2019</a>. <a href="/wiki/International_Organization_for_Standardization" title="International Organization for Standardization">International Organization for Standardization</a>. 19 May 2020. p. 4.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Standard+number+sets+and+intervals&rft.btitle=ISO+80000-2%3A2019&rft.pages=4&rft.pub=International+Organization+for+Standardization&rft.date=2020-05-19&rft_id=https%3A%2F%2Fcdn.standards.iteh.ai%2Fsamples%2F64973%2F329519100abd447ea0d49747258d1094%2FISO-80000-2-2019.pdf%23page%3D10&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoincaré1905" class="citation book cs1">Poincaré, Henri (1905) [1902]. <a class="external text" href="https://en.wikisource.org/wiki/Science_and_Hypothesis/Chapter_1">"On the nature of mathematical reasoning"</a>. La Science et l'hypothèse [Science and Hypothesis]. Translated by Greenstreet, William John. 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American Journal of Mathematics. 4 (1): 85–95. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2369151">10.2307/2369151</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2369151">2369151</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1507856">1507856</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Mathematics&rft.atitle=On+the+Logic+of+Number&rft.volume=4&rft.issue=1&rft.pages=85-95&rft.date=1881&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1507856%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2369151%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2369151&rft.aulast=Peirce&rft.aufirst=C.+S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fjstor-2369151&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShields1997" class="citation book cs1">Shields, Paul (1997). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pWjOg-zbtMAC&pg=PA43">"3. 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Indiana University Press. pp. 43–52. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-253-33020-3" title="Special:BookSources/0-253-33020-3"><bdi>0-253-33020-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=3.+Peirce%27s+Axiomatization+of+Arithmetic&rft.btitle=Studies+in+the+Logic+of+Charles+Sanders+Peirce&rft.pages=43-52&rft.pub=Indiana+University+Press&rft.date=1997&rft.isbn=0-253-33020-3&rft.aulast=Shields&rft.aufirst=Paul&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpWjOg-zbtMAC%26pg%3DPA43&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1 cs1-prop-foreign-lang-source"><a rel="nofollow" class="external text" href="https://archive.org/details/wassindundwasso00dedegoog/page/n42/mode/2up">Was sind und was sollen die Zahlen?</a> (in German). 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Boston: Addison-Wesley. p. 133. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-72634-3" title="Special:BookSources/978-0-201-72634-3"><bdi>978-0-201-72634-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+review+of+discrete+and+combinatorial+mathematics&rft.place=Boston&rft.pages=133&rft.edition=5th&rft.pub=Addison-Wesley&rft.date=2003&rft.isbn=978-0-201-72634-3&rft.aulast=Grimaldi&rft.aufirst=Ralph+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Multiplication.html">"Multiplication"</a>. mathworld.wolfram.com. Retrieved 27 July 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Multiplication&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FMultiplication.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFletcherHowell2014" class="citation book cs1">Fletcher, Harold; Howell, Arnold A. (9 May 2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7cPSBQAAQBAJ&q=Natural+numbers+closed&pg=PA116">Mathematics with Understanding</a>. Elsevier. p. 116. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4832-8079-0" title="Special:BookSources/978-1-4832-8079-0"><bdi>978-1-4832-8079-0</bdi></a>. <q>...the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+with+Understanding&rft.pages=116&rft.pub=Elsevier&rft.date=2014-05-09&rft.isbn=978-1-4832-8079-0&rft.aulast=Fletcher&rft.aufirst=Harold&rft.au=Howell%2C+Arnold+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7cPSBQAAQBAJ%26q%3DNatural%2Bnumbers%2Bclosed%26pg%3DPA116&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavisson1910" class="citation book cs1">Davisson, Schuyler Colfax (1910). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=E7oZAAAAYAAJ&q=Natural+numbers+associative&pg=PA2">College Algebra</a>. Macmillian Company. p. 2. <q>Addition of natural numbers is associative.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=College+Algebra&rft.pages=2&rft.pub=Macmillian+Company&rft.date=1910&rft.aulast=Davisson&rft.aufirst=Schuyler+Colfax&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DE7oZAAAAYAAJ%26q%3DNatural%2Bnumbers%2Bassociative%26pg%3DPA2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrandonBrownGundlachCooke1962" class="citation book cs1">Brandon, Bertha (M.); Brown, Kenneth E.; Gundlach, Bernard H.; Cooke, Ralph J. (1962). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xERMAQAAIAAJ&q=Natural+numbers+commutative">Laidlaw mathematics series</a>. Vol. 8. Laidlaw Bros. p. 25.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Laidlaw+mathematics+series&rft.pages=25&rft.pub=Laidlaw+Bros.&rft.date=1962&rft.aulast=Brandon&rft.aufirst=Bertha+%28M.%29&rft.au=Brown%2C+Kenneth+E.&rft.au=Gundlach%2C+Bernard+H.&rft.au=Cooke%2C+Ralph+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxERMAQAAIAAJ%26q%3DNatural%2Bnumbers%2Bcommutative&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-61"><a href="#cite_ref-61">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFletcherHrbacekKanoveiKatz2017" class="citation journal cs1">Fletcher, Peter; Hrbacek, Karel; Kanovei, Vladimir; Katz, Mikhail G.; Lobry, Claude; Sanders, Sam (2017). <a rel="nofollow" class="external text" href="https://doi.org/10.14321%2Frealanalexch.42.2.0193">"Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others"</a>. Real Analysis Exchange. 42 (2): 193–253. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1703.00425">1703.00425</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.14321%2Frealanalexch.42.2.0193">10.14321/realanalexch.42.2.0193</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Real+Analysis+Exchange&rft.atitle=Approaches+To+Analysis+With+Infinitesimals+Following+Robinson%2C+Nelson%2C+And+Others&rft.volume=42&rft.issue=2&rft.pages=193-253&rft.date=2017&rft_id=info%3Aarxiv%2F1703.00425&rft_id=info%3Adoi%2F10.14321%2Frealanalexch.42.2.0193&rft.aulast=Fletcher&rft.aufirst=Peter&rft.au=Hrbacek%2C+Karel&rft.au=Kanovei%2C+Vladimir&rft.au=Katz%2C+Mikhail+G.&rft.au=Lobry%2C+Claude&rft.au=Sanders%2C+Sam&rft_id=https%3A%2F%2Fdoi.org%2F10.14321%252Frealanalexch.42.2.0193&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-62"><a href="#cite_ref-62">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMints" class="citation encyclopaedia cs1">Mints, G.E. (ed.). <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php/Peano_axioms">"Peano axioms"</a>. Encyclopedia of Mathematics. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, in cooperation with the <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">European Mathematical Society</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141013163028/http://www.encyclopediaofmath.org/index.php/Peano_axioms">Archived</a> from the original on 13 October 2014. Retrieved 8 October 2014.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Peano+axioms&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=Springer%2C+in+cooperation+with+the+European+Mathematical+Society&rft_id=http%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%2FPeano_axioms&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> </li> <li id="cite_note-vonNeumann1923pp199-208-65"><a href="#cite_ref-vonNeumann1923pp199-208_65-0">^</a> <a href="#CITEREFvon_Neumann1923">von Neumann (1923)</a> </li> <li id="cite_note-Levy-66">^ <a href="#cite_ref-Levy_66-0">a</a> <a href="#cite_ref-Levy_66-1">b</a> <a href="#CITEREFLevy1979">Levy (1979)</a>, p. 52 </li> </ol></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"><h2 id="Bibliography">Bibliography</h2> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=20" title="Edit section: Bibliography" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-9 collapsible-block" id="mf-section-9"> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 25em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBluman2010" class="citation book cs1">Bluman, Allan (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=oRLc9r4bmSgC">Pre-Algebra DeMYSTiFieD</a> (Second ed.). McGraw-Hill Professional. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-174251-1" title="Special:BookSources/978-0-07-174251-1"><bdi>978-0-07-174251-1</bdi></a> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pre-Algebra+DeMYSTiFieD&rft.edition=Second&rft.pub=McGraw-Hill+Professional&rft.date=2010&rft.isbn=978-0-07-174251-1&rft.aulast=Bluman&rft.aufirst=Allan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DoRLc9r4bmSgC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarothers2000" class="citation book cs1">Carothers, N.L. (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4VFDVy1NFiAC&q=%22natural+numbers%22">Real Analysis</a>. 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Translated by Beman, Wooster Woodruff (reprint ed.). 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Translated by Beman, Wooster Woodruff. Chicago, IL: Open Court Publishing Company. Retrieved 13 August 2020 – via Project Gutenberg.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essays+on+the+Theory+of+Numbers&rft.place=Chicago%2C+IL&rft.pub=Open+Court+Publishing+Company&rft.date=1901&rft.aulast=Dedekind&rft.aufirst=Richard&rft_id=https%3A%2F%2Fwww.gutenberg.org%2Febooks%2F21016&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDedekind2007" class="citation book cs1"><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Dedekind, Richard</a> (2007) [1901]. Essays on the Theory of Numbers. 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Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90092-6" title="Special:BookSources/978-0-387-90092-6"><bdi>978-0-387-90092-6</bdi></a> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Naive+Set+Theory&rft.pub=Springer+Science+%26+Business+Media&rft.date=1960&rft.isbn=978-0-387-90092-6&rft.aulast=Halmos&rft.aufirst=Paul&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dx6cZBQ9qtgoC%26q%3Dpeano%2Baxioms%26pg%3DPP1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamilton1988" class="citation book cs1">Hamilton, A.G. (1988). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TO098EjWT38C&q=peano%27s+postulates">Logic for Mathematicians</a> (Revised ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-36865-0" title="Special:BookSources/978-0-521-36865-0"><bdi>978-0-521-36865-0</bdi></a> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logic+for+Mathematicians&rft.edition=Revised&rft.pub=Cambridge+University+Press&rft.date=1988&rft.isbn=978-0-521-36865-0&rft.aulast=Hamilton&rft.aufirst=A.G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTO098EjWT38C%26q%3Dpeano%2527s%2Bpostulates&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJamesJames1992" class="citation book cs1"><a href="/wiki/Robert_C._James" title="Robert C. James">James, Robert C.</a>; James, Glenn (1992). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UyIfgBIwLMQC">Mathematics Dictionary</a> (Fifth ed.). Chapman & Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-412-99041-0" title="Special:BookSources/978-0-412-99041-0"><bdi>978-0-412-99041-0</bdi></a> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+Dictionary&rft.edition=Fifth&rft.pub=Chapman+%26+Hall&rft.date=1992&rft.isbn=978-0-412-99041-0&rft.aulast=James&rft.aufirst=Robert+C.&rft.au=James%2C+Glenn&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DUyIfgBIwLMQC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandau1966" class="citation book cs1"><a href="/wiki/Edmund_Landau" title="Edmund Landau">Landau, Edmund</a> (1966). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DvIJBAAAQBAJ">Foundations of Analysis</a> (Third ed.). Chelsea Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-2693-5" title="Special:BookSources/978-0-8218-2693-5"><bdi>978-0-8218-2693-5</bdi></a> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Analysis&rft.edition=Third&rft.pub=Chelsea+Publishing&rft.date=1966&rft.isbn=978-0-8218-2693-5&rft.aulast=Landau&rft.aufirst=Edmund&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDvIJBAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLevy1979" class="citation book cs1"><a href="/wiki/Azriel_Levy" class="mw-redirect" title="Azriel Levy">Levy, Azriel</a> (1979). Basic Set Theory. Springer-Verlag Berlin Heidelberg. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-02310-5" title="Special:BookSources/978-3-662-02310-5"><bdi>978-3-662-02310-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Set+Theory&rft.pub=Springer-Verlag+Berlin+Heidelberg&rft.date=1979&rft.isbn=978-3-662-02310-5&rft.aulast=Levy&rft.aufirst=Azriel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMac_LaneBirkhoff1999" class="citation book cs1"><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Mac Lane, Saunders</a>; <a href="/wiki/Garrett_Birkhoff" title="Garrett Birkhoff">Birkhoff, Garrett</a> (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=L6FENd8GHIUC&q=%22the+natural+numbers%22&pg=PA15">Algebra</a> (3rd ed.). American Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-1646-2" title="Special:BookSources/978-0-8218-1646-2"><bdi>978-0-8218-1646-2</bdi></a> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.edition=3rd&rft.pub=American+Mathematical+Society&rft.date=1999&rft.isbn=978-0-8218-1646-2&rft.aulast=Mac+Lane&rft.aufirst=Saunders&rft.au=Birkhoff%2C+Garrett&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DL6FENd8GHIUC%26q%3D%2522the%2Bnatural%2Bnumbers%2522%26pg%3DPA15&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMendelson2008" class="citation book cs1"><a href="/wiki/Elliott_Mendelson" title="Elliott Mendelson">Mendelson, Elliott</a> (2008) [1973]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3domViIV7HMC">Number Systems and the Foundations of Analysis</a>. Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-45792-5" title="Special:BookSources/978-0-486-45792-5"><bdi>978-0-486-45792-5</bdi></a> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Systems+and+the+Foundations+of+Analysis&rft.pub=Dover+Publications&rft.date=2008&rft.isbn=978-0-486-45792-5&rft.aulast=Mendelson&rft.aufirst=Elliott&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3domViIV7HMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorash1991" class="citation book cs1">Morash, Ronald P. (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fH9YAAAAYAAJ">Bridge to Abstract Mathematics: Mathematical proof and structures</a> (Second ed.). Mcgraw-Hill College. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-043043-3" title="Special:BookSources/978-0-07-043043-3"><bdi>978-0-07-043043-3</bdi></a> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bridge+to+Abstract+Mathematics%3A+Mathematical+proof+and+structures&rft.edition=Second&rft.pub=Mcgraw-Hill+College&rft.date=1991&rft.isbn=978-0-07-043043-3&rft.aulast=Morash&rft.aufirst=Ronald+P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfH9YAAAAYAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMusserPetersonBurger2013" class="citation book cs1">Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=b3dbAgAAQBAJ">Mathematics for Elementary Teachers: A contemporary approach</a> (10th ed.). <a href="/wiki/Wiley_Global_Education" class="mw-redirect" title="Wiley Global Education">Wiley Global Education</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-45744-3" title="Special:BookSources/978-1-118-45744-3"><bdi>978-1-118-45744-3</bdi></a> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+for+Elementary+Teachers%3A+A+contemporary+approach&rft.edition=10th&rft.pub=Wiley+Global+Education&rft.date=2013&rft.isbn=978-1-118-45744-3&rft.aulast=Musser&rft.aufirst=Gary+L.&rft.au=Peterson%2C+Blake+E.&rft.au=Burger%2C+William+F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Db3dbAgAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSzczepanskiKositsky2008" class="citation book cs1">Szczepanski, Amy F.; Kositsky, Andrew P. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wLA2tlR_LYYC">The Complete Idiot's Guide to Pre-algebra</a>. Penguin Group. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-59257-772-9" title="Special:BookSources/978-1-59257-772-9"><bdi>978-1-59257-772-9</bdi></a> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Complete+Idiot%27s+Guide+to+Pre-algebra&rft.pub=Penguin+Group&rft.date=2008&rft.isbn=978-1-59257-772-9&rft.aulast=Szczepanski&rft.aufirst=Amy+F.&rft.au=Kositsky%2C+Andrew+P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DwLA2tlR_LYYC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomsonBrucknerBruckner2008" class="citation book cs1">Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vA9d57GxCKgC">Elementary Real Analysis</a> (Second ed.). ClassicalRealAnalysis.com. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4348-4367-8" title="Special:BookSources/978-1-4348-4367-8"><bdi>978-1-4348-4367-8</bdi></a> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Real+Analysis&rft.edition=Second&rft.pub=ClassicalRealAnalysis.com&rft.date=2008&rft.isbn=978-1-4348-4367-8&rft.aulast=Thomson&rft.aufirst=Brian+S.&rft.au=Bruckner%2C+Judith+B.&rft.au=Bruckner%2C+Andrew+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DvA9d57GxCKgC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Neumann1923" class="citation journal cs1"><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann, John</a> (1923). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141218090535/http://acta.fyx.hu/acta/showCustomerArticle.action?id=4981&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=39716d660ae98d02&style=">"Zur Einführung der transfiniten Zahlen"</a> [On the Introduction of the Transfinite Numbers]. Acta Litterarum AC Scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio Scientiarum Mathematicarum. 1: 199–208. Archived from <a rel="nofollow" class="external text" href="http://acta.fyx.hu/acta/showCustomerArticle.action?id=4981&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=39716d660ae98d02&style=">the original</a> on 18 December 2014. Retrieved 15 September 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Litterarum+AC+Scientiarum+Ragiae+Universitatis+Hungaricae+Francisco-Josephinae%2C+Sectio+Scientiarum+Mathematicarum&rft.atitle=Zur+Einf%C3%BChrung+der+transfiniten+Zahlen&rft.volume=1&rft.pages=199-208&rft.date=1923&rft.aulast=von+Neumann&rft.aufirst=John&rft_id=http%3A%2F%2Facta.fyx.hu%2Facta%2FshowCustomerArticle.action%3Fid%3D4981%26dataObjectType%3Darticle%26returnAction%3DshowCustomerVolume%26sessionDataSetId%3D39716d660ae98d02%26style%3D&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Neumann2002" class="citation book cs1"><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann, John</a> (January 2002) [1923]. <a rel="nofollow" class="external text" href="http://www.hup.harvard.edu/catalog.php?isbn=978-0674324497">"On the introduction of transfinite numbers"</a>. In van Heijenoort, Jean (ed.). From Frege to Gödel: A source book in mathematical logic, 1879–1931 (3rd ed.). Harvard University Press. pp. 346–354. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-674-32449-7" title="Special:BookSources/978-0-674-32449-7"><bdi>978-0-674-32449-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+the+introduction+of+transfinite+numbers&rft.btitle=From+Frege+to+G%C3%B6del%3A+A+source+book+in+mathematical+logic%2C+1879%E2%80%931931&rft.pages=346-354&rft.edition=3rd&rft.pub=Harvard+University+Press&rft.date=2002-01&rft.isbn=978-0-674-32449-7&rft.aulast=von+Neumann&rft.aufirst=John&rft_id=http%3A%2F%2Fwww.hup.harvard.edu%2Fcatalog.php%3Fisbn%3D978-0674324497&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"> – English translation of <a href="#CITEREFvon_Neumann1923">von Neumann 1923</a>.</li></ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"><h2 id="External_links">External links</h2> <a role="button" href="/w/index.php?title=Natural_number&action=edit&section=21" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-10 collapsible-block" id="mf-section-10"> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output 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ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" data-file-width="1024" data-file-height="1376"></noscript><span class="lazy-image-placeholder" style="width: 30px;height: 40px;" data-src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" data-alt="" data-width="30" data-height="40" data-srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-class="mw-file-element"> </div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Natural_numbers" class="extiw" title="commons:Category:Natural numbers">Natural numbers</a>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Natural_number">"Natural number"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Natural+number&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DNatural_number&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.apronus.com/provenmath/naturalaxioms.htm">"Axioms and construction of natural numbers"</a>. apronus.com.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=apronus.com&rft.atitle=Axioms+and+construction+of+natural+numbers&rft_id=http%3A%2F%2Fwww.apronus.com%2Fprovenmath%2Fnaturalaxioms.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+number" class="Z3988"></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"></div>    </section></div> <noscript><img src="https://login.m.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&mobile=1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Natural_number&oldid=1258803522">https://en.wikipedia.org/w/index.php?title=Natural_number&oldid=1258803522</a>"</div></div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"> <a class="last-modified-bar" href="/w/index.php?title=Natural_number&action=history"> <div class="post-content last-modified-bar__content"> Last edited on 21 November 2024, at 18:38 </div> </a> <div class="post-content footer-content"> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Natuurlike_getal" title="Natuurlike getal – Afrikaans" lang="af" hreflang="af" data-title="Natuurlike getal" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Nat%C3%BCrliche_Zahl" title="Natürliche Zahl – Alemannic" lang="gsw" hreflang="gsw" data-title="Natürliche Zahl" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%A8%E1%89%B0%E1%8D%88%E1%8C%A5%E1%88%AE_%E1%89%81%E1%8C%A5%E1%88%AD" title="የተፈጥሮ ቁጥር – Amharic" lang="am" hreflang="am" data-title="የተፈጥሮ ቁጥር" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%95%E0%A5%83%E0%A4%A4%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="प्राकृतिक संख्या – Angika" lang="anp" hreflang="anp" data-title="प्राकृतिक संख्या" data-language-autonym="अंगिका" data-language-local-name="Angika" class="interlanguage-link-target">अंगिका</a></li><li class="interlanguage-link interwiki-ab mw-list-item"><a href="https://ab.wikipedia.org/wiki/%D0%98%D4%A5%D1%81%D0%B0%D0%B1%D0%B0%D1%80%D0%B0%D1%82%D3%99%D1%83_%D0%B0%D1%85%D1%8B%D4%A5%D1%85%D1%8C%D0%B0%D3%A1%D0%B0%D1%80%D0%B0" title="Иԥсабаратәу ахыԥхьаӡара – Abkhazian" lang="ab" hreflang="ab" data-title="Иԥсабаратәу ахыԥхьаӡара" data-language-autonym="Аԥсшәа" data-language-local-name="Abkhazian" class="interlanguage-link-target">Аԥсшәа</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%B7%D8%A8%D9%8A%D8%B9%D9%8A" title="عدد طبيعي – Arabic" lang="ar" hreflang="ar" data-title="عدد طبيعي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Numero_natural" title="Numero natural – Aragonese" lang="an" hreflang="an" data-title="Numero natural" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D4%B2%D5%B6%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A9%D5%AB%D6%82" title="Բնական թիւ – Western Armenian" lang="hyw" hreflang="hyw" data-title="Բնական թիւ" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target">Արեւմտահայերէն</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%B8%E0%A7%8D%E0%A6%AC%E0%A6%BE%E0%A6%AD%E0%A6%BE%E0%A7%B1%E0%A6%BF%E0%A6%95_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="স্বাভাৱিক সংখ্যা – Assamese" lang="as" hreflang="as" data-title="স্বাভাৱিক সংখ্যা" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/N%C3%BAmberu_natural" title="Númberu natural – Asturian" lang="ast" hreflang="ast" data-title="Númberu natural" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Natural_%C9%99d%C9%99dl%C9%99r" title="Natural ədədlər – Azerbaijani" lang="az" hreflang="az" data-title="Natural ədədlər" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%B7%D8%A8%DB%8C%D8%B9%DB%8C_%D8%B3%D8%A7%DB%8C%DB%8C%D9%84%D8%A7%D8%B1" title="طبیعی ساییلار – South Azerbaijani" lang="azb" hreflang="azb" data-title="طبیعی ساییلار" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A7%8D%E0%A6%AC%E0%A6%BE%E0%A6%AD%E0%A6%BE%E0%A6%AC%E0%A6%BF%E0%A6%95_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="স্বাভাবিক সংখ্যা – Bangla" lang="bn" hreflang="bn" data-title="স্বাভাবিক সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ch%C5%AB-ji%C3%A2n-s%C3%B2%CD%98" title="Chū-jiân-sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Chū-jiân-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D1%8C_%D2%BB%D0%B0%D0%BD" title="Натураль һан – Bashkir" lang="ba" hreflang="ba" data-title="Натураль һан" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Натуральны лік – Belarusian" lang="be" hreflang="be" data-title="Натуральны лік" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Натуральны лік – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Натуральны лік" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D1%81%D1%82%D0%B5%D1%81%D1%82%D0%B2%D0%B5%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Естествено число – Bulgarian" lang="bg" hreflang="bg" data-title="Естествено число" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%A2%E0%BD%84%E0%BC%8B%E0%BD%96%E0%BE%B1%E0%BD%B4%E0%BD%84%E0%BC%8B%E0%BD%82%E0%BE%B2%E0%BD%84%E0%BD%A6%E0%BC%8D" title="རང་བྱུང་གྲངས། – Tibetan" lang="bo" hreflang="bo" data-title="རང་བྱུང་གྲངས།" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetan" class="interlanguage-link-target">བོད་ཡིག</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Prirodan_broj" title="Prirodan broj – Bosnian" lang="bs" hreflang="bs" data-title="Prirodan broj" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Niver_naturel" title="Niver naturel – Breton" lang="br" hreflang="br" data-title="Niver naturel" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target">Brezhoneg</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_natural" title="Nombre natural – Catalan" lang="ca" hreflang="ca" data-title="Nombre natural" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%D1%83%D1%80%D0%BB%C4%83%D1%85_%D1%85%D0%B8%D1%81%D0%B5%D0%BF%C4%95" title="Пурлăх хисепĕ – Chuvash" lang="cv" hreflang="cv" data-title="Пурлăх хисепĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/P%C5%99irozen%C3%A9_%C4%8D%C3%ADslo" title="Přirozené číslo – Czech" lang="cs" hreflang="cs" data-title="Přirozené číslo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhif_naturiol" title="Rhif naturiol – Welsh" lang="cy" hreflang="cy" data-title="Rhif naturiol" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Naturligt_tal" title="Naturligt tal – Danish" lang="da" hreflang="da" data-title="Naturligt tal" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D8%B9%D8%A7%D8%AF%D8%A7%D8%AF_%D8%B7%D8%A8%D9%8A%D8%B9%D9%8A" title="عاداد طبيعي – Moroccan Arabic" lang="ary" hreflang="ary" data-title="عاداد طبيعي" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target">الدارجة</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Nat%C3%BCrliche_Zahl" title="Natürliche Zahl – German" lang="de" hreflang="de" data-title="Natürliche Zahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Naturaalarv" title="Naturaalarv – Estonian" lang="et" hreflang="et" data-title="Naturaalarv" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A6%CF%85%CF%83%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Φυσικός αριθμός – Greek" lang="el" hreflang="el" data-title="Φυσικός αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B3mmer_natur%C3%A8l" title="Nómmer naturèl – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nómmer naturèl" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target">Emiliàn e rumagnòl</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_natural" title="Número natural – Spanish" lang="es" hreflang="es" data-title="Número natural" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Natura_nombro" title="Natura nombro – Esperanto" lang="eo" hreflang="eo" data-title="Natura nombro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_arrunt" title="Zenbaki arrunt – Basque" lang="eu" hreflang="eu" data-title="Zenbaki arrunt" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%B7%D8%A8%DB%8C%D8%B9%DB%8C" title="عدد طبیعی – Persian" lang="fa" hreflang="fa" data-title="عدد طبیعی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Teljital" title="Teljital – Faroese" lang="fo" hreflang="fo" data-title="Teljital" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target">Føroyskt</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Entier_naturel" title="Entier naturel – French" lang="fr" hreflang="fr" data-title="Entier naturel" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Uimhir_aiceanta" title="Uimhir aiceanta – Irish" lang="ga" hreflang="ga" data-title="Uimhir aiceanta" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_natural" title="Número natural – Galician" lang="gl" hreflang="gl" data-title="Número natural" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-inh mw-list-item"><a href="https://inh.wikipedia.org/wiki/%D0%91%D0%BE%D0%BA%D1%8A%D0%BE%D0%BD%D1%86%D0%B0_%D0%B4%D0%BE%D0%BB%D0%B0_%D1%82%D0%B0%D1%8C%D1%80%D0%B0%D1%85%D1%8C" title="Бокъонца дола таьрахь – Ingush" lang="inh" hreflang="inh" data-title="Бокъонца дола таьрахь" data-language-autonym="ГӀалгӀай" data-language-local-name="Ingush" class="interlanguage-link-target">ГӀалгӀай</a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E6%95%B8" title="自然數 – Gan" lang="gan" hreflang="gan" data-title="自然數" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target">贛語</a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%AA%E0%AB%8D%E0%AA%B0%E0%AA%BE%E0%AA%95%E0%AB%83%E0%AA%A4%E0%AA%BF%E0%AA%95_%E0%AA%B8%E0%AA%82%E0%AA%96%E0%AB%8D%E0%AA%AF%E0%AA%BE%E0%AA%93" title="પ્રાકૃતિક સંખ્યાઓ – Gujarati" lang="gu" hreflang="gu" data-title="પ્રાકૃતિક સંખ્યાઓ" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target">ગુજરાતી</a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%99%D0%B8%D1%80%D1%82%D0%B8%D0%BC%D2%97%D0%B8%D0%BD_%D1%82%D0%BE%D0%B9%D0%B3" title="Йиртимҗин тойг – Kalmyk" lang="xal" hreflang="xal" data-title="Йиртимҗин тойг" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target">Хальмг</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9E%90%EC%97%B0%EC%88%98" title="자연수 – Korean" lang="ko" hreflang="ko" data-title="자연수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%B6%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A9%D5%AB%D5%BE" title="Բնական թիվ – Armenian" lang="hy" hreflang="hy" data-title="Բնական թիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%95%E0%A5%83%E0%A4%A4%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="प्राकृतिक संख्या – Hindi" lang="hi" hreflang="hi" data-title="प्राकृतिक संख्या" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hsb mw-list-item"><a href="https://hsb.wikipedia.org/wiki/P%C5%99irodna_li%C4%8Dba" title="Přirodna ličba – Upper Sorbian" lang="hsb" hreflang="hsb" data-title="Přirodna ličba" data-language-autonym="Hornjoserbsce" data-language-local-name="Upper Sorbian" class="interlanguage-link-target">Hornjoserbsce</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Prirodni_broj" title="Prirodni broj – Croatian" lang="hr" hreflang="hr" data-title="Prirodni broj" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Naturala_nombro" title="Naturala nombro – Ido" lang="io" hreflang="io" data-title="Naturala nombro" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_asli" title="Bilangan asli – Indonesian" lang="id" hreflang="id" data-title="Bilangan asli" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Numero_natural" title="Numero natural – Interlingua" lang="ia" hreflang="ia" data-title="Numero natural" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D0%BE%D0%BD_%D0%BD%D1%8B%D0%BC%C3%A6%D1%86" title="Натуралон нымæц – Ossetic" lang="os" hreflang="os" data-title="Натуралон нымæц" data-language-autonym="Ирон" data-language-local-name="Ossetic" class="interlanguage-link-target">Ирон</a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/Amanani_a-natural" title="Amanani a-natural – Xhosa" lang="xh" hreflang="xh" data-title="Amanani a-natural" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target">IsiXhosa</a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Inombolo_Yokubala" title="Inombolo Yokubala – Zulu" lang="zu" hreflang="zu" data-title="Inombolo Yokubala" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target">IsiZulu</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/N%C3%A1tt%C3%BArlegar_t%C3%B6lur" title="Náttúrlegar tölur – Icelandic" lang="is" hreflang="is" data-title="Náttúrlegar tölur" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_naturale" title="Numero naturale – Italian" lang="it" hreflang="it" data-title="Numero naturale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%98%D7%91%D7%A2%D7%99" title="מספר טבעי – Hebrew" lang="he" hreflang="he" data-title="מספר טבעי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Wilangan_asli" title="Wilangan asli – Javanese" lang="jv" hreflang="jv" data-title="Wilangan asli" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target">Jawa</a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/Pilim_%C3%B1%CA%8A%CA%8A" title="Pilim ñʊʊ – Kabiye" lang="kbp" hreflang="kbp" data-title="Pilim ñʊʊ" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target">Kabɩyɛ</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B8%E0%B3%8D%E0%B2%B5%E0%B2%BE%E0%B2%AD%E0%B2%BE%E0%B2%B5%E0%B2%BF%E0%B2%95_%E0%B2%B8%E0%B2%82%E0%B2%96%E0%B3%8D%E0%B2%AF%E0%B3%86" title="ಸ್ವಾಭಾವಿಕ ಸಂಖ್ಯೆ – Kannada" lang="kn" hreflang="kn" data-title="ಸ್ವಾಭಾವಿಕ ಸಂಖ್ಯೆ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9C%E1%83%90%E1%83%A2%E1%83%A3%E1%83%A0%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98" title="ნატურალური რიცხვი – Georgian" lang="ka" hreflang="ka" data-title="ნატურალური რიცხვი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B0%D2%93%D0%B4%D1%8B%D0%BB%D1%8B_%D1%81%D0%B0%D0%BD" title="Дағдылы сан – Kazakh" lang="kk" hreflang="kk" data-title="Дағдылы сан" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Namba_asilia" title="Namba asilia – Swahili" lang="sw" hreflang="sw" data-title="Namba asilia" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Anty%C3%A9_natir%C3%A8l" title="Antyé natirèl – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Antyé natirèl" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Hejmar%C3%AAn_xwezay%C3%AE" title="Hejmarên xwezayî – Kurdish" lang="ku" hreflang="ku" data-title="Hejmarên xwezayî" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target">Kurdî</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D0%B4%D1%8B%D0%BA_%D1%81%D0%B0%D0%BD" title="Натуралдык сан – Kyrgyz" lang="ky" hreflang="ky" data-title="Натуралдык сан" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%88%E0%BA%B3%E0%BA%99%E0%BA%A7%E0%BA%99%E0%BA%97%E0%BA%B3%E0%BA%A1%E0%BA%B0%E0%BA%8A%E0%BA%B2%E0%BA%94" title="ຈຳນວນທຳມະຊາດ – Lao" lang="lo" hreflang="lo" data-title="ຈຳນວນທຳມະຊາດ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target">ລາວ</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_naturalis" title="Numerus naturalis – Latin" lang="la" hreflang="la" data-title="Numerus naturalis" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Natur%C4%81ls_skaitlis" title="Naturāls skaitlis – Latvian" lang="lv" hreflang="lv" data-title="Naturāls skaitlis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Natierlech_Zuel" title="Natierlech Zuel – Luxembourgish" lang="lb" hreflang="lb" data-title="Natierlech Zuel" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Nat%C5%ABralusis_skai%C4%8Dius" title="Natūralusis skaičius – Lithuanian" lang="lt" hreflang="lt" data-title="Natūralusis skaičius" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Natuurlek_getal" title="Natuurlek getal – Limburgish" lang="li" hreflang="li" data-title="Natuurlek getal" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target">Limburgs</a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Numero_natural" title="Numero natural – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Numero natural" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target">Lingua Franca Nova</a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/kacna%27u" title="kacna'u – Lojban" lang="jbo" hreflang="jbo" data-title="kacna'u" data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target">La .lojban.</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/N%C3%BCmar_nat%C3%BCraal" title="Nümar natüraal – Lombard" lang="lmo" hreflang="lmo" data-title="Nümar natüraal" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Term%C3%A9szetes_sz%C3%A1mok" title="Természetes számok – Hungarian" lang="hu" hreflang="hu" data-title="Természetes számok" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D1%80%D0%BE%D0%B4%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Природен број – Macedonian" lang="mk" hreflang="mk" data-title="Природен број" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Isa_nanahary" title="Isa nanahary – Malagasy" lang="mg" hreflang="mg" data-title="Isa nanahary" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%8E%E0%B4%A3%E0%B5%8D%E0%B4%A3%E0%B5%BD_%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="എണ്ണൽ സംഖ്യ – Malayalam" lang="ml" hreflang="ml" data-title="എണ്ണൽ സംഖ്യ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Numru_naturali" title="Numru naturali – Maltese" lang="mt" hreflang="mt" data-title="Numru naturali" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target">Malti</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%A8%E0%A5%88%E0%A4%B8%E0%A4%B0%E0%A5%8D%E0%A4%97%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="नैसर्गिक संख्या – Marathi" lang="mr" hreflang="mr" data-title="नैसर्गिक संख्या" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%B7%D8%A8%D9%8A%D8%B9%D9%89" title="عدد طبيعى – Egyptian Arabic" lang="arz" hreflang="arz" data-title="عدد طبيعى" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target">مصرى</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nombor_asli" title="Nombor asli – Malay" lang="ms" hreflang="ms" data-title="Nombor asli" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/N%C3%BAmaro_natural" title="Númaro natural – Mirandese" lang="mwl" hreflang="mwl" data-title="Númaro natural" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target">Mirandés</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB_%D1%82%D0%BE%D0%BE" title="Натурал тоо – Mongolian" lang="mn" hreflang="mn" data-title="Натурал тоо" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9E%E1%80%98%E1%80%AC%E1%80%9D%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8" title="သဘာဝကိန်း – Burmese" lang="my" hreflang="my" data-title="သဘာဝကိန်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Naba_vakatamata" title="Naba vakatamata – Fijian" lang="fj" hreflang="fj" data-title="Naba vakatamata" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target">Na Vosa Vakaviti</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Natuurlijk_getal" title="Natuurlijk getal – Dutch" lang="nl" hreflang="nl" data-title="Natuurlijk getal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%95%E0%A5%83%E0%A4%A4%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%99%E0%A5%8D%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="प्राकृतिक सङ्ख्या – Nepali" lang="ne" hreflang="ne" data-title="प्राकृतिक सङ्ख्या" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target">नेपाली</a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%95%E0%A5%83%E0%A4%A4%E0%A4%BF%E0%A4%95_%E0%A4%B2%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%83" title="प्राकृतिक ल्याः – Newari" lang="new" hreflang="new" data-title="प्राकृतिक ल्याः" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target">नेपाल भाषा</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E6%95%B0" title="自然数 – Japanese" lang="ja" hreflang="ja" data-title="自然数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Nat%C3%BC%C3%BCrelk_taal" title="Natüürelk taal – Northern Frisian" lang="frr" hreflang="frr" data-title="Natüürelk taal" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Naturlig_tall" title="Naturlig tall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Naturlig tall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Naturleg_tal" title="Naturleg tal – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Naturleg tal" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Enti%C3%A8r_natural" title="Entièr natural – Occitan" lang="oc" hreflang="oc" data-title="Entièr natural" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%97%E0%AC%A3%E0%AC%A8_%E0%AC%B8%E0%AC%82%E0%AC%96%E0%AD%8D%E0%AD%9F%E0%AC%BE" title="ଗଣନ ସଂଖ୍ୟା – Odia" lang="or" hreflang="or" data-title="ଗଣନ ସଂଖ୍ୟା" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target">ଓଡ଼ିଆ</a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Lakkoofsa_lakkaawwii" title="Lakkoofsa lakkaawwii – Oromo" lang="om" hreflang="om" data-title="Lakkoofsa lakkaawwii" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target">Oromoo</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Natural_son" title="Natural son – Uzbek" lang="uz" hreflang="uz" data-title="Natural son" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%81%E0%A8%A6%E0%A8%B0%E0%A8%A4%E0%A9%80_%E0%A8%85%E0%A9%B0%E0%A8%95" title="ਕੁਦਰਤੀ ਅੰਕ – Punjabi" lang="pa" hreflang="pa" data-title="ਕੁਦਰਤੀ ਅੰਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%86%DB%8C%DA%86%D8%B1%D9%84_%D9%86%D9%85%D8%A8%D8%B1" title="نیچرل نمبر – Western Punjabi" lang="pnb" hreflang="pnb" data-title="نیچرل نمبر" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Nachral_nomba" title="Nachral nomba – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Nachral nomba" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%85%E1%9F%86%E1%9E%93%E1%9E%BD%E1%9E%93%E1%9E%82%E1%9E%8F%E1%9F%8B%E1%9E%92%E1%9E%98%E1%9F%92%E1%9E%98%E1%9E%87%E1%9E%B6%E1%9E%8F%E1%9E%B7" title="ចំនួនគត់ធម្មជាតិ – Khmer" lang="km" hreflang="km" data-title="ចំនួនគត់ធម្មជាតិ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target">ភាសាខ្មែរ</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_natural" title="Nùmer natural – Piedmontese" lang="pms" hreflang="pms" data-title="Nùmer natural" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Nat%C3%BCrliche_Tall" title="Natürliche Tall – Low German" lang="nds" hreflang="nds" data-title="Natürliche Tall" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_naturalne" title="Liczby naturalne – Polish" lang="pl" hreflang="pl" data-title="Liczby naturalne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_natural" title="Número natural – Portuguese" lang="pt" hreflang="pt" data-title="Número natural" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ksh mw-list-item"><a href="https://ksh.wikipedia.org/wiki/Nat%C3%B6rliche_Zahle" title="Natörliche Zahle – Colognian" lang="ksh" hreflang="ksh" data-title="Natörliche Zahle" data-language-autonym="Ripoarisch" data-language-local-name="Colognian" class="interlanguage-link-target">Ripoarisch</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_natural" title="Număr natural – Romanian" lang="ro" hreflang="ro" data-title="Număr natural" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Натуральное число – Russian" lang="ru" hreflang="ru" data-title="Натуральное число" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D0%B9_%D1%87%D1%8B%D1%8Bh%D1%8B%D0%BB%D0%B0%D2%95%D0%B0" title="Натуральнай чыыhылаҕа – Yakut" lang="sah" hreflang="sah" data-title="Натуральнай чыыhылаҕа" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target">Саха тыла</a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%95%E0%A5%83%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A4%B8%E0%A4%99%E0%A5%8D%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="प्राकृतिकसङ्ख्या – Sanskrit" lang="sa" hreflang="sa" data-title="प्राकृतिकसङ्ख्या" data-language-autonym="संस्कृतम्" data-language-local-name="Sanskrit" class="interlanguage-link-target">संस्कृतम्</a></li><li class="interlanguage-link interwiki-sc mw-list-item"><a href="https://sc.wikipedia.org/wiki/N%C3%B9meru_naturale" title="Nùmeru naturale – Sardinian" lang="sc" hreflang="sc" data-title="Nùmeru naturale" data-language-autonym="Sardu" data-language-local-name="Sardinian" class="interlanguage-link-target">Sardu</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numrat_natyror%C3%AB" title="Numrat natyrorë – Albanian" lang="sq" hreflang="sq" data-title="Numrat natyrorë" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_naturali" title="Nùmmuru naturali – Sicilian" lang="scn" hreflang="scn" data-title="Nùmmuru naturali" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%83%E0%B7%8A%E0%B7%80%E0%B7%8F%E0%B6%B7%E0%B7%8F%E0%B7%80%E0%B7%92%E0%B6%9A_%E0%B7%83%E0%B6%82%E0%B6%9B%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F" title="ස්වාභාවික සංඛ්‍යා – Sinhala" lang="si" hreflang="si" data-title="ස්වාභාවික සංඛ්‍යා" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Natural_number" title="Natural number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Natural number" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Prirodzen%C3%A9_%C4%8D%C3%ADslo" title="Prirodzené číslo – Slovak" lang="sk" hreflang="sk" data-title="Prirodzené číslo" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Naravno_%C5%A1tevilo" title="Naravno število – Slovenian" lang="sl" hreflang="sl" data-title="Naravno število" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Naturalno_n%C5%AFmera" title="Naturalno nůmera – Silesian" lang="szl" hreflang="szl" data-title="Naturalno nůmera" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target">Ślůnski</a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Thiin_tirsiimo" title="Thiin tirsiimo – Somali" lang="so" hreflang="so" data-title="Thiin tirsiimo" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target">Soomaaliga</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%D8%B3%D8%B1%D9%88%D8%B4%D8%AA%DB%8C" title="ژمارەی سروشتی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ژمارەی سروشتی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D1%80%D0%BE%D0%B4%D0%B0%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Природан број – Serbian" lang="sr" hreflang="sr" data-title="Природан број" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Prirodan_broj" title="Prirodan broj – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Prirodan broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Luonnollinen_luku" title="Luonnollinen luku – Finnish" lang="fi" hreflang="fi" data-title="Luonnollinen luku" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Naturliga_tal" title="Naturliga tal – Swedish" lang="sv" hreflang="sv" data-title="Naturliga tal" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Likas_na_bilang" title="Likas na bilang – Tagalog" lang="tl" hreflang="tl" data-title="Likas na bilang" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%8E%E0%AE%A3%E0%AF%8D" title="இயல் எண் – Tamil" lang="ta" hreflang="ta" data-title="இயல் எண்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A2%D0%B0%D0%B1%D0%B8%D0%B3%D1%8B%D0%B9_%D1%81%D0%B0%D0%BD" title="Табигый сан – Tatar" lang="tt" hreflang="tt" data-title="Табигый сан" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B8%E0%B0%B9%E0%B0%9C_%E0%B0%B8%E0%B0%82%E0%B0%96%E0%B1%8D%E0%B0%AF" title="సహజ సంఖ్య – Telugu" lang="te" hreflang="te" data-title="సహజ సంఖ్య" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%98%E0%B8%A3%E0%B8%A3%E0%B8%A1%E0%B8%8A%E0%B8%B2%E0%B8%95%E0%B8%B4" title="จำนวนธรรมชาติ – Thai" lang="th" hreflang="th" data-title="จำนวนธรรมชาติ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%B4%D0%B0%D0%B4%D0%B8_%D0%BD%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D3%A3" title="Адади натуралӣ – Tajik" lang="tg" hreflang="tg" data-title="Адади натуралӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Do%C4%9Fal_say%C4%B1lar" title="Doğal sayılar – Turkish" lang="tr" hreflang="tr" data-title="Doğal sayılar" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Natural_sanlar" title="Natural sanlar – Turkmen" lang="tk" hreflang="tk" data-title="Natural sanlar" data-language-autonym="Türkmençe" data-language-local-name="Turkmen" class="interlanguage-link-target">Türkmençe</a></li><li class="interlanguage-link interwiki-kcg mw-list-item"><a href="https://kcg.wikipedia.org/wiki/A%CC%B1za%CC%B1za%CC%B1rak_la%CC%B1mba" title="A̱za̱za̱rak la̱mba – Tyap" lang="kcg" hreflang="kcg" data-title="A̱za̱za̱rak la̱mba" data-language-autonym="Tyap" data-language-local-name="Tyap" class="interlanguage-link-target">Tyap</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Натуральне число – Ukrainian" lang="uk" hreflang="uk" data-title="Натуральне число" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%82%D8%AF%D8%B1%D8%AA%DB%8C_%D8%B9%D8%AF%D8%AF" title="قدرتی عدد – Urdu" lang="ur" hreflang="ur" data-title="قدرتی عدد" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_t%E1%BB%B1_nhi%C3%AAn" title="Số tự nhiên – Vietnamese" lang="vi" hreflang="vi" data-title="Số tự nhiên" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/T%C3%BCk%C3%BCarv" title="Tüküarv – Võro" lang="vro" hreflang="vro" data-title="Tüküarv" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target">Võro</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E6%95%B8" title="自然數 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="自然數" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Natuurlik_getal" title="Natuurlik getal – West Flemish" lang="vls" hreflang="vls" data-title="Natuurlik getal" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target">West-Vlams</a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Unob_nga_ihap" title="Unob nga ihap – Waray" lang="war" hreflang="war" data-title="Unob nga ihap" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target">Winaray</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E6%95%B0" title="自然数 – Wu" lang="wuu" hreflang="wuu" data-title="自然数" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A0%D7%90%D7%98%D7%99%D7%A8%D7%9C%D7%A2%D7%9B%D7%A2_%D7%A6%D7%90%D7%9C" title="נאטירלעכע צאל – Yiddish" lang="yi" hreflang="yi" data-title="נאטירלעכע צאל" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target">ייִדיש</a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/N%E1%BB%8D%CC%81mb%C3%A0_%C3%A0d%C3%A1b%C3%A1y%C3%A9" title="Nọ́mbà àdábáyé – Yoruba" lang="yo" hreflang="yo" data-title="Nọ́mbà àdábáyé" data-language-autonym="Yorùbá" data-language-local-name="Yoruba" class="interlanguage-link-target">Yorùbá</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E6%95%B8" title="自然數 – Cantonese" lang="yue" hreflang="yue" data-title="自然數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Nat%C5%ABral%C4%97%CC%84j%C4%97_skaitl%C4%93" title="Natūralė̄jė skaitlē – Samogitian" lang="sgs" hreflang="sgs" data-title="Natūralė̄jė skaitlē" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target">Žemaitėška</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E6%95%B0" title="自然数 – Chinese" lang="zh" hreflang="zh" data-title="自然数" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li><li class="interlanguage-link interwiki-zgh mw-list-item"><a href="https://zgh.wikipedia.org/wiki/%E2%B5%93%E2%B5%8E%E2%B5%8E%E2%B5%89%E2%B4%B7_%E2%B4%B0%E2%B4%B3%E2%B4%B0%E2%B5%8E%E2%B4%B0%E2%B5%8F" title="ⵓⵎⵎⵉⴷ ⴰⴳⴰⵎⴰⵏ – Standard Moroccan Tamazight" lang="zgh" hreflang="zgh" data-title="ⵓⵎⵎⵉⴷ ⴰⴳⴰⵎⴰⵏ" data-language-autonym="ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ" data-language-local-name="Standard Moroccan Tamazight" class="interlanguage-link-target">ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ</a></li></ul> </section> </div> <div class="minerva-footer-logo"><img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"/> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod"> This page was last edited on 21 November 2024, at 18:38 (UTC).</li> <li id="footer-info-copyright">Content is available under 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<script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-7l5xc","wgBackendResponseTime":234,"wgPageParseReport":{"limitreport":{"cputime":"1.288","walltime":"1.657","ppvisitednodes":{"value":12406,"limit":1000000},"postexpandincludesize":{"value":286832,"limit":2097152},"templateargumentsize":{"value":12768,"limit":2097152},"expansiondepth":{"value":13,"limit":100},"expensivefunctioncount":{"value":7,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":273304,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1307.151 1 -total"," 27.69% 361.902 2 Template:Reflist"," 21.61% 282.483 50 Template:Cite_book"," 9.06% 118.382 104 Template:Math"," 7.59% 99.179 1 Template:Lang"," 7.45% 97.424 1 Template:Short_description"," 7.37% 96.326 8 Template:Navbox"," 7.10% 92.807 8 Template:Efn"," 6.45% 84.340 1 Template:Number_systems"," 5.41% 70.699 8 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