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In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Natural numbers are sometimes called whole numbers, a term that may also refer to all integers, including the negative ones. Natural numbers are also called sometimes counting numbers, particularly in primary education. The set of the natural numbers is commonly denoted with a bold or a blackboard bold .

The natural numbers are used for counting, and for labeling the result of a count, like "there are seven days in a week", in which case they are called cardinal numeral. They are also used to label places in an ordered series, like "the third day of the month", in which case they are called ordinal numeral. Natural numbers may also be used to label, like the jersey numbers of a sports team; in this case, they have no specific mathematical properties and are called .

Two natural operations are defined on natural numbers, addition and multiplication. Arithmetic is the study of the ways to perform these operations. 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.

Many are built from the natural numbers and contain them. For example, the are made by including 0 and negative numbers. The add fractions, and the add all infinite decimals. add the Imaginary unit. says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers." This makes up natural numbers as foundational for all mathematics.: "Numbers make up the foundation of mathematics."

Intuitive concept

An intuitive and implicit understanding of natural numbers is developed naturally through using numbers for counting, ordering and basic arithmetic. Within this are two closely related aspects of what a natural number is: the size of a collection; 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?".

(2025). 9780810106055, Northwestern Univ. Press. ISBN 9780810106055

A natural number used in this way describes a characteristic of a collection of objects. This characteristic, the size of a collection is called cardinality and a natural number used to describe or measure it is called a cardinal number. Two collections have the same size or cardinality if there is a one-to-one correspondence between the objects in each collection to the objects in the other. For example, in the image to the right every apple can be paired off with one orange and every orange can be paired off with one apple. From this, even without counting or using numbers it can be seen that the group of apples has the same cardinality as the group of oranges, meaning they are both assigned the same cardinal number.

The natural number 3 is the thing used for the particular cardinal number described above and for the cardinal number of any other collection of objects that could be paired off in the same way to one of these groups.

Position in a sequence

The natural numbers have a fixed progression, which is the familiar sequence beginning with 1, 2, 3, and so on. A natural number can be used to denote a specific position in any other sequence, in which case it is called an ordinal number To have a specific position in a sequence means to come either before or after every other position in the sequence in a defined way, which is the concept of Order theory.

The natural number 3 then is the thing that comes after 2 and 1, and before 4, 5 and so on. The number 2 is the thing that comes after 1, and 1 is the first element in the sequence. Each number represents the relation that position bears to the rest of the infinite sequence.

Counting

The process of counting involves both the cardinal and ordinal use of the natural numbers and illustrates the way the two fit together. To count the number of objects in a collection, each object is paired off with a natural number, usually by mentally or verbally saying the name of the number and assigning it to a particular object. The numbers must be assigned in order starting with 1 (they are ordinal) but the order of the objects chosen is arbitrary as long as each object is assigned one and only one number. When all of the objects have been assigned a number, the ordinal number assigned to the final object gives the result of the count, which is the cardinal number of the whole collection.

History

Ancient roots

The most primitive method of representing a natural number is to use one's fingers, as in finger counting. Putting down a tally mark 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 numeral system to represent numbers. This allowed systems to be developed for recording large numbers. The ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, and all powers of 10 up to over 1 million. A stone carving from Karnak, dating back from around 1500 BCE and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The had a place-value 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.

Georges Ifrah (2025). 9780471375685, Wiley. ISBN 9780471375685

A much later advance was the development of the idea that can be considered as a number, with its own numeral. The use of a 0 numerical digit 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. The Olmec and Maya civilizations used 0 as a separate number as early as the , but this usage did not spread beyond Mesoamerica.

Charles C. Mann (2025). 9781400040063, Knopf. . ISBN 9781400040063

Brian Evans (2025). 9781118853979, John Wiley & Sons. ISBN 9781118853979

The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628 CE. However, 0 had been used as a number in the medieval computus (the calculation of the date of Easter), beginning with Dionysius Exiguus in 525 CE, without being denoted by a numeral. Standard Roman numerals 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.

The first systematic study of numbers as is usually credited to the ancient Greece philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all. Euclid, 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).

Ian Mueller (2025). 9780486453002, Dover Publications. ISBN 9780486453002

However, in the definition of perfect number which comes shortly afterward, Euclid treats 1 as a number like any other. In definition VII.3 a "part" was defined as a number, but here 1 is considered to be a part, so that for example is a perfect number.

Independent studies on numbers also occurred at around the same time in India, China, and Mesoamerica.

Morris Kline (1990). 9780195061352, Oxford University Press. ISBN 9780195061352

Emergence as a term

Nicolas Chuquet used the term progression naturelle (natural progression) in 1484. The earliest known use of "natural number" as a complete English phrase is in 1763. The 1771 Encyclopaedia Britannica defines natural numbers in the logarithm article.

Starting at 0 or 1 has long been a matter of definition. In 1727, Bernard Le Bovier de Fontenelle wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0. In 1889, Giuseppe Peano used N for the positive integers and started at 1, but he later changed to using N₀ and N₁. Historically, most definitions have excluded 0, but many mathematicians such as George A. Wentworth, Bertrand Russell, Nicolas Bourbaki, Paul Halmos, Stephen Cole Kleene, and John Horton Conway have preferred to include 0.

Mathematicians have noted tendencies in which definition is used, such as algebra texts including 0, number theory and analysis texts excluding 0,

(2021). 9783030838997, Springer Nature. . ISBN 9783030838997

See, for example, or logic and set theory texts including 0,

(2025). 9780691118802, Princeton university press. ISBN 9780691118802

(1998). 9780412606106, Chapman & Hall/CRC. . ISBN 9780412606106

dictionaries excluding 0, school books (through high-school level) excluding 0, and upper-division college-level books including 0.

(1977). 9780122384400, Academic Press. ISBN 9780122384400

There are exceptions to each of these tendencies and as of 2023 no formal survey has been conducted. Arguments raised include division by zero and the size of the empty set. Computer languages often start from zero when enumerating items like For loop and string- or array-elements. Including 0 began to rise in popularity in the 1960s. The ISO 31-11 standard included 0 in the natural numbers in its first edition in 1978 and this has continued through its present edition as ISO 80000-2.

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. Leopold Kronecker summarized his belief as "God made the integers, all else is the work of man".

The constructivists saw a need to improve upon the logical rigor in the foundations of mathematics. 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 Gottlob 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.

In 1881, Charles Sanders Peirce provided the first axiomatization of natural-number arithmetic.

Paul Shields (1997). 9780253330208, Indiana University Press. . ISBN 9780253330208

In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of Dedekind's axioms in his book The principles of arithmetic presented by a new method (). This approach is now called Peano arithmetic. It is based on an axiomatization of the properties of : 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. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem.

Notation

The set of all natural numbers is standardly denoted or

\mathbb N.

Older texts have occasionally employed as the symbol for this set.

W. Rudin (1976). 9780070542358, McGraw-Hill. . ISBN 9780070542358

Since natural numbers may contain 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, possibly with reference to the Examples include:

Naturals without zero ( $\{1,2,...\}$ ): $\mathbb{N}^*$ , $\mathbb{N}_{>0}$ , $\mathbb{Z}_{>0}$ , $\mathbb{Z}^{\ge 1}$ , $\mathbb{N}^{\ge 1}$ ,
(2025). 9783736980730, Cuvillier Verlag. ISBN 9783736980730
$\mathbb Z^+$ ,
(2025). 9780201726343, Pearson Addison Wesley. ISBN 9780201726343
$\mathbb N^+$ ,
(2019). 9781108488068, Cambridge University Press. . ISBN 9781108488068
$\mathbb{N}_1$ .
Naturals with zero ( $\{0,1,2,...\}$ ): $\mathbb{Z}^{\ge 0}$ , $\mathbb{N}^{\ge 0}$ , $\mathbb{N}_0$ ,
(2015). 9780191016486, OUP Oxford. . ISBN 9780191016486
$\mathbb N^0$ ,
(2025). 9788120841680, Motilal Banarsidass. . ISBN 9788120841680
$\mathbb Z^{0+}$ .
(2022). 9781800611825, World Scientific. . ISBN 9781800611825

Definitions

There are two standard methods for formally defining natural numbers. The first one is called Peano arithmetic and consists of an autonomous axiomatic theory based on a few axioms called the Peano axioms, named for Giuseppe Peano. The second definition is based on set theory and defines the natural numbers as specific sets. More precisely, each natural number is defined as an explicitly defined set, whose elements allow counting the elements of other sets, in the sense that the sentence "a set has elements" means that there exists a one to one correspondence between the two sets and .

The sets used to define natural numbers satisfy the Peano axioms. It follows that every theorem 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 Fermat's Last Theorem.

The definition of the integers as sets satisfying the Peano axioms provide a model of Peano arithmetic inside set theory. An important consequence is that, if set theory is consistent (as is usually assumed), then Peano arithmetic is consistent. In other words, if a contradiction could be proved in Peano arithmetic, then set theory would be logically inconsistent.

Peano axioms

The five Peano axioms are the following:

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 $x$ equals the successor of $y$ , then $x$ equals $y$ .
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 $x$ is $x + 1$ .

Set-theoretic definition

Intuitively, the natural number is the common property of all sets that have elements. So, it seems natural to define as an equivalence class under the relation "can be made in one to one correspondence". This does not work in all set theory, as such an equivalence class would not be a set (because of Russell's paradox). The standard solution is to define a particular set with elements that will be called the natural number .

The following definition was first published by John von Neumann, although Levy attributes the idea to unpublished work of Zermelo in 1916. As this definition extends to infinite set as a definition of ordinal number, the sets considered below are sometimes called von Neumann ordinals.

The definition proceeds as follows:

Call , the empty set.
Define the successor of any set by .
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.

It follows that the natural numbers are defined iteratively as follows:

* ,

* etc.

It can be checked that the natural numbers satisfy the Peano axioms.

With this definition, given a natural number , the sentence "a set has elements" can be formally defined as "there exists a bijection from to ." This formalizes the operation of counting the elements of . Also, if and only if is a subset of . In other words, the set inclusion defines the usual total order on the natural numbers. This order is a well-order.

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 von Neumann definition of ordinals for defining all , including the infinite ones: "each ordinal is the well-ordered set of all smaller ordinals."

If one finitism, 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, Ernst Zermelo provided a construction that is nowadays only of historical interest, and is sometimes referred to as . It consists in defining as the empty set, and .

With this definition each nonzero natural number is a singleton set. So, the property of the natural numbers to represent cardinalities is not directly accessible; only the ordinal property (being the th element of a sequence) is immediate. Unlike von Neumann's construction, the Zermelo ordinals do not extend to infinite ordinals.

Properties

This section uses the convention

\mathbb{N}=\mathbb{N}_0=\mathbb{N}^*\cup\{0\}

Addition

Given the set

\mathbb{N}

of natural numbers and the successor function

S \colon \mathbb{N} \to \mathbb{N}

sending each natural number to the next one, one can define addition of natural numbers recursively by setting and for all , . Thus, , , and so on. The algebraic structure

(\mathbb{N}, +)

is a commutative monoid with identity element 0. It is a free object 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 .

If 1 is defined as , then . That is, is simply the successor of .

Multiplication

Analogously, given that addition has been defined, a multiplication operator

\times

can be defined via and . This turns

(\mathbb{N}^*, \times)

into a free commutative monoid with identity element 1; a generator set for this monoid is the set of .

Relationship between addition and multiplication

Addition and multiplication are compatible, which is expressed in the distributivity: . These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that

\mathbb{N}

is not closed under subtraction (that is, subtracting one natural from another does not always result in another natural), means that

\mathbb{N}

is not a ring; instead it is a semiring (also known as a rig).

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

Order

In this section, juxtaposed variables such as indicate the product , and the standard order of operations is assumed.

A total order on the natural numbers is defined by letting if and only if there exists another natural number where . This order is compatible with the arithmetical operations in the following sense: if , and are natural numbers and , then and .

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

Division

In this section, juxtaposed variables such as indicate the product , and the standard order of operations is assumed.

While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder or Euclidean division is available as a substitute: for any two natural numbers and with there are natural numbers and such that

a = bq + r \text{ and } r < b.

The number is called the quotient and is called the remainder of the division of by . The numbers and are uniquely determined by and . 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 and , both and are natural numbers.
(2014). 9781483280790, Elsevier. . ISBN 9781483280790
Associativity: for all natural numbers , , and , and .
Commutativity: for all natural numbers and , and .
Existence of : for every natural number , and .

If the natural numbers are taken as "excluding 0", and "starting at 1", then for every natural number , . However, the "existence of additive identity element" property is not satisfied

Distributivity of multiplication over addition for all natural numbers , , and , .
No nonzero : if and are natural numbers such that , then or (or both).

Generalizations

Natural numbers are broadly used in two ways: to quantify and to order. A number used to represent the quantity of objects in a collection ("there are 6 coins on the table") is called a cardinal numeral, while a number used to order individual objects within a collection ("she finished 6th in the race") is an ordinal numeral.

These two uses of natural numbers apply only to . Georg Cantor discovered at the end of the 19th century that both uses of natural numbers can be generalized to , but that they lead to two different concepts of "infinite" numbers, the and the .

Other generalizations of natural numbers are discussed in .

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