{{Short description|Number used for counting}}{{pp-vandalism|small=yes}} {{Use dmy dates|date=May 2021}} right|thumb|upright|Natural numbers can be used for counting: one apple plus two apples equals three apples.
In mathematics, the '''natural numbers''' are the numbers 0, 1, 2, 3, and so on, possibly excluding 0.{{efn|It depends on authors and context whether 0 is considered a natural number.}}<ref name="Enderton">{{cite book |last1=Enderton |first1=Herbert B. |title=Elements of set theory |date=1977 |publisher=Academic Press |location=New York |isbn=0122384407 |page=66}}</ref> The terms '''positive integers''', '''non-negative integers''', '''whole numbers''', and '''counting numbers''' are also used.<ref name="Cooke2000">{{cite book |last1=Cooke |first1=Heather |url=https://books.google.com/books?id=yNCL5iTJqLQC&dq=%22counting%20numbers%22%20%22positive%20integers%22&pg=PA14 |title=Primary Mathematics |date=26 October 2000 |publisher=SAGE |isbn=978-1-84787-949-3 |page=14 |language=en}}</ref><ref name="Zegarelli2014">{{cite book |last1=Zegarelli |first1=Mark |url=https://books.google.com/books?id=V8nDAgAAQBAJ&dq=%22counting%20numbers%22&pg=PA21 |title=Basic Math and Pre-Algebra For Dummies |date=28 January 2014 |publisher=John Wiley & Sons |isbn=978-1-118-79199-8 |page=21 |language=en |quote=Counting numbers (also called natural numbers): The set of numbers beginning 1, 2, 3, 4, ... and going on infinitely.}}</ref> The set of the natural numbers is commonly denoted by a bold {{math|'''N'''}} or a blackboard bold {{tmath|\N}}.
The natural numbers are used for counting,<!-- Please, do not link this word that is used in its common language meaning, and not in any technical meaning --> and for labeling the result of a count, such as: "there are ''seven'' days in a week", in which case they are called ''cardinal numbers''. They are also used to label places in an ordered series,<!-- Please, do not link these words that are used in their common language meanings, and not in any technical meaning --> such as: "the ''third'' day of the month", in which case they are called ''ordinal numbers''.<ref>{{Cite book |last=Tao |first=Terence|author-link=Terence Tao |url=http://link.springer.com/10.1007/978-981-10-1789-6 |title=Analysis I |date=2016 |publisher=Springer Singapore |isbn=978-981-10-1789-6 |series=Texts and Readings in Mathematics |volume=37 |location=Singapore |page=68 |doi=10.1007/978-981-10-1789-6 }}</ref>
Natural numbers are commonly expressed in writing using ten symbols called numerals ("0 1 2 3 4 5 6 7 8 9"). These numerals can also be used as unique identifiers or labels (like the jersey numbers of a sports team) that are referred to as ''nominal numbers,''<ref>{{cite journal |last1=Woodin |first1=Greg |first2=Bodo |last2=Winter |title=Numbers in Context: Cardinals, Ordinals, and Nominals in American English |journal=Cognitive Science |volume=48|number=6 |year=2024 |article-number=e13471 |doi=10.1111/cogs.13471 |doi-access=free|pmid=38895756 |pmc=11475258 }}</ref> which resemble natural numbers but have no specific mathematical properties.
Natural numbers can be compared by magnitude, with larger numbers coming after smaller ones in the list 1, 2, 3, .... Two basic arithmetical operations are defined on natural numbers: addition and multiplication. However, the inverse operations, subtraction and division, only sometimes give natural-number results: subtracting a larger natural number from a smaller one results in a negative number and dividing one natural number by another commonly leaves a remainder.
The most common number systems used throughout mathematics – the integers, rational numbers, real numbers, and complex numbers – contain the natural numbers, and can be formally defined in terms of natural numbers.<ref>{{harvtxt|Mendelson|2008|page=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."</ref><ref>{{harvtxt|Bluman|2010|page=1}}: "Numbers make up the foundation of mathematics."</ref>
Arithmetic is the study of the ways to perform basic operations on these number systems. Number theory is the study of the properties of these operations and their generalizations. Much of combinatorics involves counting mathematical objects, patterns and structures that are defined using natural numbers.
==Intuitive concept== An intuitive and implicit understanding of natural numbers is developed ''naturally'' through using numbers for counting, ordering and basic arithmetic.<ref>{{Harvard citation year brackets|Mayberry|2000|p=xvi|ps=: "We gain our knowledge of these numbers when we learn to count them out and to calculate with them, so we are led to see these processes of counting out and calculating as constitutive of the very notion of natural number."}}</ref> Within this are two closely related aspects of what a natural number is: the ''size of a collection;''<ref>{{Harvard citation year brackets|Tao|2016|p=68|ps=: says, "...one of our main conceptualizations of natural numbers (is) that of cardinality, or measuring how many elements there are in a set".}}</ref> and ''a'' ''position in a sequence''.
=== Size of a collection === Natural numbers can be used to answer questions like "how many apples are on the table?"<ref>{{Cite book |last1=Frege |first1=Gottlob |title=The foundations of arithmetic: a logico-mathematical enquiry into the concept of number |last2=Frege |first2=Gottlob |orig-date=1953|date=1975 |publisher=Northwestern Univ. Press |isbn=978-0-8101-0605-5 |edition=2. revised |location=Evanston Ill |pages=5}}</ref> A natural number used in this way describes a characteristic of a ''finite collection of objects''. This characteristic, the ''size of a collection'', is called cardinality and a natural number used to describe or measure it is a cardinal number. alt=Left: A group of three apples. Right: A group of three oranges.|thumb|A group of apples and group of oranges with the same cardinality Two finite collections have the same size or cardinality if they have a one-to-one correspondence, meaning the objects can be arranged in pairs (one from each collection), with every object in exactly one pair. In the adjacent image every apple is paired with exactly one orange and every orange is paired with exactly one apple. As such, the group of apples has the ''same'' cardinality as the group of oranges, or put more simply the number of apples is the same as the number of oranges.
Because this equality can be established without counting or using any prior notion of number,<ref>{{Harvard citation year brackets|Mayberry|2000|p=135|ps=: "Cantor's discovery was that ...'sameness of size', defined in terms of one-to-one correspondence, is logically prior to the notion of counting or, indeed, to the notion of 'number'..."}}</ref><ref>{{Harvard citation year brackets|Russell|2012|ps=: "In actual fact, it is simpler logically to find out whether two collections have the same number of terms than it is to define what that number is."|p=12}}</ref> it can form the ''definition'' of a cardinal number.<ref>{{Harvard citation year brackets|Russell|2012|p=14|ps=: "It is very easy to see that if (for example) a collection has three members, the class of all those collections that are similar to it will be the class of trios. And whatever number of terms a collection may have, those collections that are “similar” to it will have the same number of terms. We may take this as a definition of “having the same number of terms.”}}</ref> In this case, the number of apples, oranges - and of any other collection that could be paired off to either group - is 3.
If two collections do not have the same cardinality, pairing will leave one of the collections with objects that are unpaired and this can be used to define a size relationship between them. The collection in which all objects are paired is said to be "smaller" and the one left with unpaired objects "larger", than the other.
=== Position in a sequence === A sequence is a list of objects in a specific order. More precisely, a sequence is a function that assigns an object to each position in that list. The positions themselves are labeled using a well-ordered set; every element always has a clear next element.<ref>{{cite book |last1=Abbott |first1=James Crawford |title=Sets, lattices, and Boolean algebras |date=1969 |page=87 |quote=Our final concept is a generalization of the notion of a sequence as defined above for the natural numbers, in which we replace <math>\N</math> by an arbitrary well-ordered set <math>\langle A, \leq \rangle</math>. If <math>X</math> is any set and <math>I</math> is an ideal of <math>A</math>, then a function <math>s_I</math> defined on <math>I</math> with values in <math>X</math> is called a sequence of type <math>I</math>. It is therefore a subset of <math>X</math> indexed by elements of <math>I</math>; i.e., <math>s_i = \{\langle x, s(x) \rangle \mid x \in I\}</math>.}}</ref> Every well-ordered set has an order type, which is the ordinal number that describes its shape of ordering.<ref>{{cite book |last1=Jech |first1=Thomas J. |title=Set theory |date=2003 |publisher=Springer |location=Berlin ; New York |isbn=978-3540440857 |page=20 |edition=The 3rd millennium, rev. and expanded |quote=Theorem 2.12. Every well-ordered set is isomorphic to a unique ordinal number.}}</ref> The position labels here are not counts or size like with the cardinal numbers, just ordered elements.<ref>{{Harvard citation year brackets|Quine|1960|p=|ps= says the only thing required of an acceptable description of natural numbers is that they form a progression and so "any progression—i.e., any infinite series each of whose members has only finitely many precursors—will do nicely".|pp=262-263}}</ref>
The natural numbers are the most common choice for labeling infinite sequences because they form the simplest infinite well-ordered set, with order type ω. They start at either 0 or 1 and continue in their familiar fixed order — 1, 2, 3, and so on — with no end point. Each natural number labels a specific position in the sequence based on where it falls relative to all other positions. For example, 1 is the first position, 2 is the position right after 1, and 3 is the position after both 1 and 2 and before 4, 5, and so on. This ordering matches the usual ordering, smaller numbers before larger ones. But the natural numbers are simply the most familiar example; any well-ordered set would work equally well for indexing a sequence, for example the set of letters a, b, c, and so on.<ref>{{Harvard citation year brackets|Benacerraf|1965|p=70|ps=: "To be the number 3 is no more and no less than to be preceded by 2, 1, and possibly 0, and to be followed by 4,5, and so forth...Any object can play the role of 3; that is, any object can be the third element in some progression. What is peculiar to 3 is that it defines that role - not by being a paradigm of any object which plays it, but by representing the relation that any third member of a progression bears to the rest of the progression."}}</ref>
== Terminology and notation ==
The term ''natural numbers'' has two common definitions: either {{math|1=0, 1, 2, ...}} or {{math|1, 2, 3, ...}}. Because there is no universal convention, the definition can be chosen to suit the context of use.<ref name="Enderton" /><ref name=":1">{{cite web |last=Weisstein |first=Eric W. |title=Natural Number |url=https://mathworld.wolfram.com/NaturalNumber.html |access-date=11 August 2020 |website=mathworld.wolfram.com |language=en}}</ref> To eliminate ambiguity, the sequences {{math|1, 2, 3, ...}} and {{math|1=0, 1, 2, ...}} are often called the '''positive integers''' and the '''non-negative integers''', respectively.
The phrase '''whole numbers''' is frequently used for the natural numbers that include 0, although it may also mean all integers, positive and negative.<ref>{{cite dictionary |year=2003 |title=integer |dictionary=Embedded Systems Dictionary |publisher=Taylor & Francis |url=https://books.google.com/books?id=zePGx82d_fwC |access-date=28 March 2017 |pages=138 (integer), 247 (signed integer), & 276 (unsigned integer) |isbn=978-1-57820-120-4 |url-status=live |archive-url=https://web.archive.org/web/20170329150719/https://books.google.com/books?id=zePGx82d_fwC |archive-date=29 March 2017 |last2=Barr |first2=Michael |name-list-style=amp |first1=Jack G. |last1=Ganssle |via=Google Books}}</ref><ref name="Cooke2000" /> In primary education, '''counting numbers''' usually refer to the natural numbers starting at 1,<ref name="Zegarelli2014" /> though this definition can vary.<ref>{{cite book |last1=Rice |first1=Harris |chapter-url=https://books.google.com/books?id=2kwH8pZKLLEC&dq=%22counting%20numbers%22&pg=PA393 |title=The Mathematics Teacher |date=1922 |publisher=National Council of Teachers of Mathematics |page=393 |language=en |chapter=Errors in computations and the rounded number |quote=A counting number is the number given in answer to the question "How many?" In this class of numbers belongs zero and positive integers/}}</ref><ref name="MathWorld_CountingNumber">{{MathWorld|title=Counting Number|id=CountingNumber}}</ref>
The set of all natural numbers is typically denoted {{math|'''N'''}} or in blackboard bold as <math>\mathbb N.</math><ref name=":1" /><ref>{{cite web |title=Listing of the Mathematical Notations used in the Mathematical Functions Website: Numbers, variables, and functions |url=https://functions.wolfram.com/Notations/1/ |access-date=27 July 2020 |website=functions.wolfram.com}}</ref>{{Efn|Older texts have occasionally employed {{math|''J''}} as the symbol for this set.<ref>{{cite book |url=https://archive.org/details/1979RudinW |title=Principles of Mathematical Analysis |last=Rudin |first=W. |publisher=McGraw-Hill |year=1976 |isbn=978-0-07-054235-8 |location=New York |page=25}}</ref>}} Whether 0 is included is often determined by the context but may also be specified by using <math>\mathbb N</math> or <math>\mathbb Z</math> (the set of all integers) with a subscript or superscript. Examples include <math>\mathbb{N}_1</math>,<ref name="Peano19012">{{cite book |last1=Peano |first1=Giuseppe |url=https://archive.org/details/formulairedesmat00pean/page/38/mode/2up |title=Formulaire des mathematiques |date=1901 |publisher=Paris, Gauthier-Villars |page=39 |language=fr}}</ref> or <math>\mathbb Z^+</math><ref name="Grimaldi20042">{{cite book |last1=Grimaldi |first1=Ralph P. |title=Discrete and Combinatorial Mathematics: An applied introduction |publisher=Pearson Addison Wesley |year=2004 |isbn=978-0-201-72634-3 |edition=5th}}</ref> (for the set starting at 1) and <math>\mathbb{N}_0</math><ref>{{cite book |last1=Stewart |first1=Ian |url=https://books.google.com/books?id=xSN-BwAAQBAJ&dq=natural%20numbers%20%22N0%22&pg=PA160 |title=The Foundations of Mathematics |last2=Tall |first2=David |date=12 March 2015 |publisher=OUP Oxford |isbn=978-0-19-101648-6 |page=160 |language=en |access-date=30 July 2025}}</ref> or <math>\mathbb Z^{0+}</math><ref>{{cite book |last1=Fokas |first1=Athanassios |url=https://books.google.com/books?id=QwuhEAAAQBAJ&dq=natural%20numbers%20%22z0%2B%22&pg=PA4 |title=Modern Mathematical Methods For Scientists And Engineers: A Street-smart Introduction |last2=Kaxiras |first2=Efthimios |date=12 December 2022 |publisher=World Scientific |isbn=978-1-80061-182-5 |page=4 |language=en |access-date=30 July 2025 |author-link2=Efthimios Kaxiras}}</ref> (for the set including 0).
=== Numeral === A '''numeral''' is a symbol or grouping of symbols used to express a natural number in writing, and a particular set of symbols with specific rules for using them is a numeral system. Each symbol in a numeral system represents a unique natural number - said to be its ''value'' - and can be used alone as a numeral, or in a string with other symbols which together form a numeral.
The decimal system which uses Arabic numerals and positional notation rules is the universal standard for representing natural numbers in mathematics and in common use. In part because of this universal standard, the distinction between an abstract number (a value) and its symbol (a numeral) is generally unimportant so numerals are frequently referred to simply as "numbers". This is sometimes done even where the distinction is relevant, as with binary numerals which are often called "binary numbers".
== Use of natural numbers == Natural numbers are used for counting and the four basic operations of arithmetic: addition, subtraction, multiplication, and division.
=== Counting === thumb|The cardinality principle of counting Counting is the process of iterating through the natural numbers in sequential order starting at 1. It can be done using numbers alone (as in "counting ''to 10''"), or by applying the count to objects (as in "counting ''the students in the class''").
When applied to a collection of objects, counting determines the cardinality of the collection by establishing a one-to-one correspondence between the objects and the natural numbers.<ref>{{Harvard citation year brackets|Russell|2012|p=13}}</ref> This involves consecutively "tagging" each object with a number while maintaining a running partition of the tagged objects from those not yet tagged.<ref>{{Harvard citation year brackets|Gelman|Gallistel|1986}}</ref> The numbers must be assigned in order starting at 1 - so they are ordinal numbers - but the order of the objects chosen is arbitrary as long as each object receives one and only one number. The '''cardinality principle''' is the understanding that the ordinal number assigned to the final object gives the result of the count: the cardinal number of the collection.<ref>{{Harvard citation year brackets|Carey|2009|p=289|ps=: "In counting, the symbols must be applied in order, in 1–1 correspondence to the individuals in the set being enumerated (1–1 correspondence principle). The cardinal value of the set is determined by the ordinal position of the last symbol reached in the count (cardinality principle)."}}</ref>
==Formal definitions== Formal definitions take the existing, intuitive notion of natural numbers together with the rules of arithmetic and define them both in the more fundamental terms of mathematical logic. Formal systems typically assume that the defining characteristic of natural numbers is their fixed order<ref>{{Harvard citation year brackets|Tao|2016|p=68|ps=: "Indeed, the Peano axiom approach treats natural numbers more like ordinals than cardinals."}}</ref><ref>{{Cite book |last=Russell |first=Bertrand |url=https://www.gutenberg.org/ebooks/41654 |title=Introduction to Mathematical Philosophy |date=2012-12-18 |language=English}}</ref> and establish this order using the primitive notion of a successor. Every natural number has a successor, which is another unique natural number that it is followed by.
Two standard formal definitions are based on the Peano axioms and set theory. The Peano axioms (named for Giuseppe Peano) do not explicitly define what the natural numbers ''are'', but instead comprise a list of statements or axioms that must be true of natural numbers, however they are defined. In contrast, set theory defines each natural number as a particular set, in which a set can be generally understood as a collection of distinct objects or elements. While the two methods are different, they are consistent in that the natural number sets collectively ''satisfy'' the Peano axioms.
===Peano axioms=== {{Main|Peano axioms}}
The five Peano axioms are:<ref>{{cite encyclopedia |editor-first=G.E. |editor-last=Mints |title=Peano axioms |encyclopedia=Encyclopedia of Mathematics |publisher=Springer, in cooperation with the European Mathematical Society |url=http://www.encyclopediaofmath.org/index.php/Peano_axioms |url-status=live |access-date=8 October 2014 |archive-url=https://web.archive.org/web/20141013163028/http://www.encyclopediaofmath.org/index.php/Peano_axioms |archive-date=13 October 2014 }}</ref>{{efn|{{harvtxt|Hamilton|1988|pages=117 ff}} calls them "Peano's Postulates" and begins with "1.{{spaces|2}}0 is a natural number."<br/> {{harvtxt|Halmos|1974|page=46}} uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I){{spaces|2}}{{math|0 ∈ ω}} (where, of course, {{math|0 {{=}} ∅}}" ({{math|ω}} is the set of all natural numbers).<br/> {{harvtxt|Morash|1991}} gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1: ''An Axiomatization for the System of Positive Integers'') }}
# 0 is a natural number. # Every natural number has a successor which is also a natural number. # 0 is not the successor of any natural number. # If the successor of <math> x </math> equals the successor of <math> y </math>, then <math> x</math> equals <math> y</math>. # The axiom of induction: 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.
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> x</math> is <math> x + 1</math>.
===Set-theoretic definition=== {{Main|Set-theoretic definition of natural numbers|von Neumann ordinal}}
In set theory each natural number {{mvar|n}} is defined as a specific set. A variety of constructions have been proposed, however the standard solution (due to John von Neumann)<ref name="vonNeumann1923pp199-208">{{Harvp|von Neumann|1923}}</ref> is: * Call {{math|0 {{=}} {{mset| }}}}, the empty set. * Define the successor {{math|''S''(''a'')}} of any set {{mvar|a}} by {{math|''S''(''a'') {{=}} ''a'' ∪ {{mset|''a''}}}}. * By the axiom of infinity, there exist sets which contain 0 and are closed under the successor function. Such sets are said to be ''inductive''. The intersection of all inductive sets is still an inductive set. * This intersection is the set of the ''natural numbers''.
This produces an iterative definition of the natural numbers called the von Neumann ordinals:
<math>\begin{alignat}{2} 0 & && {} = \{\} && {} = \varnothing \\ 1 & = 0 \cup \{0\} && {} = \{0\} && = \{ \varnothing\} \\ 2 & = 1 \cup \{1\} && {} = \{0,1\} && {} = \{\varnothing,\{\varnothing\}\}\\ 3 & = 2 \cup \{2\} && {} = \{0,1,2\} && {} = \{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}\} \\ n & = n-1 \cup \{n-1\} && {} = \{0,1, ...,n-1\} && {} = \{\varnothing,\{\varnothing\},...,\{\varnothing,\{\varnothing\},...\}\}\\ \end{alignat}</math>
In this construction every natural number {{mvar|n}} is a set containing {{mvar|n}} elements, where each element is a natural number less than {{mvar|n}}. From this, the intuitive concepts of cardinality and order can be formally defined as:
* Cardinality: a set {{mvar|S}} has {{mvar|n}} elements if there is a one-to-one correspondence or bijection from {{mvar|n}} to {{mvar|S}}. * Order: {{math|''n'' ≤ ''m''}} if and only if {{math|''n''}} is a subset of {{math|''m''}}.
Another construction sometimes called '''{{vanchor|Zermelo ordinals}}'''<ref name="Levy">{{harvp|Levy|1979|page=52}}</ref> defines {{math|0 {{=}} {{mset| }}}} and {{math|''S''(''a'') {{=}} {{mset|''a''}}}} and is now largely only of historical interest.
==Properties== This section uses the convention that 0 is a natural number: <math>\mathbb{N}=\mathbb{N}_0</math>.
===Addition=== Given the set <math>\mathbb{N}</math> of natural numbers and the successor function <math>S \colon \mathbb{N} \to \mathbb{N}</math> sending each natural number to the next one, addition (<math>+</math>) is defined by:<blockquote><math>\begin{align} a + 0 & = a & \textrm{(1)}\\ a + S(b) & = S(a+b) & \textrm{(2)}\\ \end{align}</math></blockquote>In the statements above, (1) explicitly defines addition for the first natural number and (2) gives a recursive definition for each subsequent number in terms of previous definitions, as illustrated below.<blockquote><math>\begin{alignat}{2} & a + 1 = a + S(0) = S(a+0) = S(a) \\ & a + 2 = a + S(1) = S(a+1) = S(S(a)) \\ & a + 3 = a + S(2) = S(a+2) = S(S(S(a))) \end{alignat}</math></blockquote>In this way, addition can be seen as repeated application of the successor function. Intuitively, {{math|''a'' + ''b''}} is evaluated by applying the successor function to {{math|''a''}} as many times as it must be applied to {{math|0}} to produce {{math|''b''}}.
The algebraic structure <math>(\mathbb{N}, +)</math> is a commutative monoid with identity element 0. It is a free monoid on one generator. This commutative monoid satisfies the cancellation property, so it can be embedded in a group. The smallest group containing the natural numbers is the integers.
===Multiplication=== Analogously, given that addition has been defined, a multiplication operator <math>\times</math> can be defined via {{math|''a'' × 0 {{=}} 0}} and {{math|''a'' × S(''b'') {{=}} (''a'' × ''b'') + ''a''}}. This turns <math>(\mathbb{N}^*, \times)</math> into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers.
===Relationship between addition and multiplication=== In the natural numbers, addition and multiplication are compatible, which is expressed in the distribution law: {{math|''a'' × (''b'' + ''c'') {{=}} (''a'' × ''b'') + (''a'' × ''c'')}}. However <math>\mathbb{N}</math> is not closed under subtraction (that is, subtracting one natural from another does not always result in another natural); equivalently, one can say that <math>\mathbb{N}</math> lacks additive inverses. These properties of addition and multiplication mean that <math>\mathbb{N}</math> is ''not'' a ring; instead it is a semiring (also known as a ''rig''). Semirings are an algebraic generalization of rings where multiplication is not necessarily commutative, although multiplication in <math>\mathbb{N}</math> is commutative.
If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin with {{math|''a'' + 1 {{=}} ''S''(''a'')}} and {{math|''a'' × 1 {{=}} ''a''}}. Furthermore, <math>(\mathbb{N^*}, +)</math> has no identity element.
===Order===
A total order on the natural numbers is defined by letting {{math|''a'' ≤ ''b''}} if and only if there exists another natural number {{math|''c''}} where {{math|''a'' + ''c'' {{=}} ''b''}}. This order is compatible with the arithmetical operations in the following sense: if {{math|''a''}}, {{math|''b''}} and {{math|''c''}} are natural numbers and {{math|''a'' ≤ ''b''}}, then {{math|''a'' + ''c'' ≤ ''b'' + ''c''}} and {{math|''a'' × ''c'' ≤ ''b'' × ''c''}}.
An important property of the natural numbers is that they are well-ordered: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an ordinal number; for the natural numbers, this is denoted as {{math|''ω''}} (omega).
===Division=== While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of ''division with remainder'' or Euclidean division is available as a substitute: for any two natural numbers {{math|''a''}} and {{math|''b''}} with {{math|''b'' ≠ 0}} there are natural numbers {{math|''q''}} and {{math|''r''}} such that :<math>a = b \times q + r \text{ and } r < b. </math>
The number {{math|''q''}} is called the ''quotient'' and {{math|''r''}} is called the ''remainder'' of the division of {{math|''a''}} by {{math|''b''}}. The numbers {{math|''q''}} and {{math|''r''}} are uniquely determined by {{math|''a''}} and {{math|''b''}}. This Euclidean division is key to the several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.
===Algebraic properties satisfied by the natural numbers=== The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties: * Closure under addition and multiplication: for all natural numbers {{math|''a''}} and {{math|''b''}}, both {{math|''a'' + ''b''}} and {{math|''a'' × ''b''}} are natural numbers.<ref>{{cite book |last1=Fletcher |first1=Harold |last2=Howell |first2=Arnold A. |date=9 May 2014 |title=Mathematics with Understanding |publisher=Elsevier |isbn=978-1-4832-8079-0 |page=116 |language=en |url=https://books.google.com/books?id=7cPSBQAAQBAJ&q=Natural+numbers+closed&pg=PA116 |quote=...the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication}}</ref> * Associativity: for all natural numbers {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}, {{math|''a'' + (''b'' + ''c'') {{=}} (''a'' + ''b'') + ''c''}} and {{math|''a'' × (''b'' × ''c'') {{=}} (''a'' × ''b'') × ''c''}}.<ref>{{cite book |last=Davisson |first=Schuyler Colfax |title=College Algebra |date=1910 |publisher=Macmillian Company |page=2 |language=en |url=https://books.google.com/books?id=E7oZAAAAYAAJ&q=Natural+numbers+associative&pg=PA2 |quote=Addition of natural numbers is associative.}}</ref> * Commutativity: for all natural numbers {{math|''a''}} and {{math|''b''}}, {{math|''a'' + ''b'' {{=}} ''b'' + ''a''}} and {{math|''a'' × ''b'' {{=}} ''b'' × ''a''}}.<ref>{{cite book |last1=Brandon |first1=Bertha (M.) |last2=Brown |first2=Kenneth E. |last3=Gundlach |first3=Bernard H. |last4=Cooke |first4=Ralph J. |date=1962 |title=Laidlaw mathematics series |publisher=Laidlaw Bros. |volume=8 |page=25 |language=en |url=https://books.google.com/books?id=xERMAQAAIAAJ&q=Natural+numbers+commutative}}</ref> * Existence of identity elements: for every natural number {{Math|''a''}}, {{math|''a'' + 0 {{=}} ''a''}} and {{math|''a'' × 1 {{=}} ''a''}}. ** If the natural numbers are taken as "excluding 0", and "starting at 1", then for every natural number {{Math|''a''}}, {{math|''a'' × 1 {{=}} ''a''}}. However, the "existence of additive identity element" property is not satisfied * Distributivity of multiplication over addition for all natural numbers {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}, {{math|''a'' × (''b'' + ''c'') {{=}} (''a'' × ''b'') + (''a'' × ''c'')}}. * No nonzero zero divisors: if {{math|''a''}} and {{math|''b''}} are natural numbers such that {{math|''a'' × ''b'' {{=}} 0}}, then {{math|''a'' {{=}} 0}} or {{math|''b'' {{=}} 0}} (or both).
==History== For most of history, what are now called natural numbers were simply ''numbers''. Between the late middle ages and end of the 17th century, the concept of number expanded to include negative, rational and irrational numbers, becoming what we now call the real numbers.<ref>{{Harvard citation year brackets|Mayberry|2000|p=17}}</ref> With this came the need to distinguish between the original numbers and these new types.<ref>{{Cite book |last=Ifrah |first=Georges |url=https://books.google.com.au/books/about/The_Universal_History_of_Numbers.html?id=FMTI7rwevZcC&redir_esc=y |title=The Universal History of Numbers: From Prehistory to the Invention of the Computer |date=2000-10-09 |publisher=Wiley |isbn=978-0-471-39340-5 |language=en}}</ref>
Nicolas Chuquet used the term ''progression naturelle'' (natural progression) in 1484.<ref>{{cite book |last1=Chuquet |first1=Nicolas |author-link=Nicolas Chuquet |url=https://gallica.bnf.fr/ark:/12148/bpt6k62599266/f75.image |title=Le Triparty en la science des nombres |date=1881 |language=fr |orig-date=1484}}</ref> The earliest known use of "natural number" as a complete English phrase is in 1763.<ref>{{cite book |last1=Emerson |first1=William |url=https://archive.org/details/bim_eighteenth-century_the-method-of-increments_emerson-william_1763/page/112/mode/2up |title=The method of increments |date=1763 |page=113}}</ref><ref name="MacTutor">{{cite web |title=Earliest Known Uses of Some of the Words of Mathematics (N) |url=https://mathshistory.st-andrews.ac.uk/Miller/mathword/n/ |website=Maths History |language=en}}</ref> The 1771 Encyclopaedia Britannica defines natural numbers in the logarithm article.<ref name="MacTutor" />
===Formal construction===
In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. Henri Poincaré 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.<ref>{{cite book |last1=Poincaré |first1=Henri |title=La Science et l'hypothèse |date=1905 |at=VI |translator1-last=Greenstreet |translator1-first=William John |trans-title=Science and Hypothesis |chapter=On the nature of mathematical reasoning |orig-date=1902 |chapter-url=https://en.wikisource.org/wiki/Science_and_Hypothesis/Chapter_1}}</ref> Leopold Kronecker summarized his belief as "God made the integers, all else is the work of man".{{efn|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."<ref>{{cite book |last=Gray |first=Jeremy |author-link=Jeremy Gray (mathematician) |year=2008 |title=Plato's Ghost: The modernist transformation of mathematics |page=153 |publisher=Princeton University Press |isbn=978-1-4008-2904-0 |via=Google Books |url=https://books.google.com/books?id=ldzseiuZbsIC&q=%22God+made+the+integers%2C+all+else+is+the+work+of+man.%22 |url-status=live |archive-url=https://web.archive.org/web/20170329150904/https://books.google.com/books?id=ldzseiuZbsIC&q=%22God+made+the+integers%2C+all+else+is+the+work+of+man.%22#v=snippet&q=%22God%20made%20the%20integers%2C%20all%20else%20is%20the%20work%20of%20man.%22&f=false |archive-date=29 March 2017 }}</ref><ref>{{cite book |last=Weber |first=Heinrich L. |year=1891–1892 |chapter=Kronecker |chapter-url=http://www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6 |archive-url=https://web.archive.org/web/20180809110042/http://www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6 |archive-date=9 August 2018 |title=''Jahresbericht der Deutschen Mathematiker-Vereinigung'' |trans-title=Annual report of the German Mathematicians Association |pages=2:5–23. (The quote is on p. 19) |postscript=; }} {{cite web |title=access to ''Jahresbericht der Deutschen Mathematiker-Vereinigung'' |url=http://www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002 |archive-url=https://web.archive.org/web/20170820201100/http://www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002 |archive-date=20 August 2017 }}</ref>}}
The constructivists saw a need to improve upon the logical rigor in the foundations of mathematics.{{efn|"Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." {{harv|Eves|1990|p=606}} }} In the 1860s, Hermann Grassmann suggested a recursive definition 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.
Set-theoretical definitions of natural numbers were initiated by Frege. 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 Russell's paradox. 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.<ref>{{harvnb|Eves|1990|loc=Chapter 15}}</ref>
In 1881, Charles Sanders Peirce provided the first axiomatization of natural-number arithmetic.<ref>{{cite journal |last=Peirce |first=C. S. |author-link=Charles Sanders Peirce |year=1881 |title=On the Logic of Number |url=https://archive.org/details/jstor-2369151 |journal=American Journal of Mathematics |volume=4 |issue=1 |pages=85–95 |doi=10.2307/2369151 |jstor=2369151 |mr=1507856}}</ref><ref>{{cite book |last=Shields |first=Paul |url=https://archive.org/details/studiesinlogicof00nath |title=Studies in the Logic of Charles Sanders Peirce |publisher=Indiana University Press |year=1997 |isbn=0-253-33020-3 |editor1-last=Houser |editor1-first=Nathan |pages=43–52 |chapter=3. Peirce's Axiomatization of Arithmetic |editor2-last=Roberts |editor2-first=Don D. |editor3-last=Van Evra |editor3-first=James |chapter-url=https://books.google.com/books?id=pWjOg-zbtMAC&pg=PA43 |url-access=registration}}</ref> In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic,<ref>{{cite book |url=https://archive.org/details/wassindundwasso00dedegoog/page/n42/mode/2up |title=Was sind und was sollen die Zahlen? |date=1893 |publisher=F. Vieweg |at=71–73 |language=German}}</ref> and in 1889, Peano published a simplified version of Dedekind's axioms in his book ''The principles of arithmetic presented by a new method'' ({{langx|la|Arithmetices principia, nova methodo exposita}}). This approach is now called Peano arithmetic. It is based on an axiomatization of the properties of ordinal numbers: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several weak systems of set theory. One such system is ZFC with the axiom of infinity replaced by its negation.<ref>{{cite journal |last1=Baratella |first1=Stefano |last2=Ferro |first2=Ruggero |year=1993 |title=A theory of sets with the negation of the axiom of infinity |journal=Mathematical Logic Quarterly |volume=39 |issue=3 |pages=338–352 |doi=10.1002/malq.19930390138 |mr=1270381}}</ref> Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem.<ref>{{cite journal |last1=Kirby |first1=Laurie |last2=Paris |first2=Jeff |year=1982 |title=Accessible Independence Results for Peano Arithmetic |journal=Bulletin of the London Mathematical Society |publisher=Wiley |volume=14 |issue=4 |pages=285–293 |doi=10.1112/blms/14.4.285 |issn=0024-6093}}</ref>
=== Zero as natural number === Starting at 0 or 1 has long been a matter of convention. In 1727, Bernard Le Bovier de Fontenelle argued both ways, that 0 could be a ''term'' as in a sequence 0, 1, 2, ..., but that 1 was a basic ''element'' out of which other numbers could be formed by repeated addition.<ref>{{cite book |last1=Fontenelle |first1=Bernard de |url=https://gallica.bnf.fr/ark:/12148/bpt6k64762n/f31.item |title=Eléments de la géométrie de l'infini |date=1727 |page=3 |language=fr}}</ref> In 1889, Giuseppe Peano used N for the positive integers and started at 1,<ref>{{cite book |url=https://archive.org/details/arithmeticespri00peangoog/page/n12/mode/2up |title=Arithmetices principia: nova methodo |date=1889 |publisher=Fratres Bocca |page=12 |language=Latin}}</ref> but he later changed to using N<sub>0</sub> and N<sub>1</sub>.<ref name="Peano1901">{{cite book |last1=Peano |first1=Giuseppe |url=https://archive.org/details/formulairedesmat00pean/page/38/mode/2up |title=Formulaire des mathematiques |date=1901 |publisher=Paris, Gauthier-Villars |page=39 |language=fr}}</ref> Most early authors excluded 0,<ref name="MacTutor" /><ref>{{cite book |last1=Fine |first1=Henry Burchard |url=https://books.google.com/books?id=RR4PAAAAIAAJ&dq=%22natural%20number%22&pg=PA6 |title=A College Algebra |date=1904 |publisher=Ginn |page=6 |language=en}}</ref><ref>{{cite book |url=https://books.google.com/books?id=184i06Py1ZYC&dq=%22natural%20number%22%201&pg=PA12 |title=Advanced Algebra: A Study Guide to be Used with USAFI Course MC 166 Or CC166 |date=1958 |publisher=United States Armed Forces Institute |page=12 |language=en}}</ref> but many mathematicians such as George A. Wentworth, Bertrand Russell, Nicolas Bourbaki, Paul Halmos, Stephen Cole Kleene, and John Horton Conway included 0.<ref>{{cite web |title=Natural Number |url=https://archive.lib.msu.edu/crcmath/math/math/n/n035.htm |website=archive.lib.msu.edu}}</ref><ref name="MacTutor" /> Including 0 gained wider adoption in the 1960s<ref name="MacTutor" /> and was formalized in ISO 31-11 (1978), which defines natural numbers to include 0, a convention retained in the current ISO 80000-2 standard.<ref name="ISO80000">{{cite book |url=https://www.iso.org/standard/64973.html |title=ISO 80000-2:2019 Quantities and units Part 2: Mathematics |date=24 June 2025 |publisher=International Organization for Standardization |chapter=Standard number sets and intervals |chapter-url=https://cdn.standards.iteh.ai/samples/64973/329519100abd447ea0d49747258d1094/ISO-80000-2-2019.pdf#page=10}}</ref>
==Generalizations== {{Classification of numbers}}The most common number systems used throughout mathematics are extensions of the natural numbers, in the sense that each of them contains a subset which has the same arithmetical structure. These number systems can also be formally defined in terms of natural numbers (though they need not be{{efn|The commonly-assumed set-theoretic containment may be obtained by constructing the reals, discarding any earlier constructions, and defining the other sets as subsets of the final construction.<ref>{{cite book |last1=Wohlgemuth |first1=Andrew |title=Introduction to Proof in Abstract Mathematics |date=10 June 2014 |publisher=Courier Corporation |isbn=978-0-486-14168-8 |page=237 |url=https://books.google.com/books?id=PEP_AwAAQBAJ&pg=PA237 |language=en}}</ref>}}). If the difference of every two natural numbers is considered to be a number, the result is the integers, which include zero and negative numbers. If the quotient of every two integers is considered to be a number, the result is the rational numbers, including fractions. If every infinite decimal is considered to be a number, the result is the real numbers. If every solution of a polynomial equation is considered to be a number, the result is the complex numbers.
Other generalizations of natural numbers are discussed in {{section link|Number#Extensions of the concept}}.
==See also== {{Portal|Mathematics}} * {{annotated link|Canonical representation of a positive integer}} * {{annotated link|Countable set}} * Sequence – Function of the natural numbers in another set * {{annotated link|Ordinal number}} * {{annotated link|Cardinal number}} * {{annotated link|Set-theoretic definition of natural numbers}} ==Notes== {{Notelist}}
==References== {{Reflist|25em}}
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==External links== {{Commons category|Natural numbers}} * {{springer|title=Natural number|id=p/n066090}} * {{cite web |title=Axioms and construction of natural numbers |website=apronus.com |url=http://www.apronus.com/provenmath/naturalaxioms.htm }}
{{Number systems}} {{Classes of natural numbers}} {{Authority control}}
Category:Cardinal numbers Category:Elementary mathematics Category:Integers Category:Number theory Category:Sets of real numbers