Example of a natural number: 6. There are 6 apples in this picture and 6 is shown as an Arabic numeral.

In mathematics, the natural numbers are the numbers 1, 2, 3, etc., possibly including 0 as well. Some definitions, including the standard ISO 80000-2,[1] begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, ..., whereas others start with 1, corresponding to the positive integers 1, 2, 3, ...[2][lower-alpha 1] Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers).[4] In common language, particularly in primary school education, natural numbers may be called counting numbers[5] to intuitively exclude the negative integers and zero, and also to contrast the discreteness of counting to the continuity of measurement—a hallmark characteristic of real numbers.

The natural numbers can be used for counting (as in "there are six coins on the table"), in which case they serve as cardinal numbers. They may also be used for ordering (as in "this is the third largest city in the country"), in which case they serve as ordinal numbers. Natural numbers are sometimes used as labels—also known as nominal numbers, (e.g. jersey numbers in sports)—which do not have the properties of numbers in a mathematical sense.[3][6]

The natural numbers form a set, often symbolized as . Many other number sets are built by successively extending the set of natural numbers: the integers, by including an additive identity 0 (if not yet in) and an additive inverse −n for each nonzero natural number n; the rational numbers, by including a multiplicative inverse for each nonzero integer n (and also the product of these inverses by integers); the real numbers by including the limits of Cauchy sequences[lower-alpha 2] of rationals; the complex numbers, by adjoining to the real numbers a square root of −1 (and also the sums and products thereof); and so on.[lower-alpha 3][lower-alpha 4] This chain of extensions canonically embeds the natural numbers in the other number systems.

Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics.

History

Ancient roots

The Ishango bone (on exhibition at the Royal Belgian Institute of Natural Sciences)[7][8][9] is believed to have been used 20,000 years ago for natural number arithmetic.

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 numerals 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 Babylonians 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.[10]

A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 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.[lower-alpha 5] The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BCE, but this usage did not spread beyond Mesoamerica.[12][13] 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.[14]

The first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.[lower-alpha 6] 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).[16] However, in the definition of perfect number which comes shortly afterward, Euclid treats 1 as a number like any other.[17]

Independent studies on numbers also occurred at around the same time in India, China, and Mesoamerica.[18]

Modern definitions

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.[19] Leopold Kronecker summarized his belief as "God made the integers, all else is the work of man".[lower-alpha 7]

The constructivists saw a need to improve upon the logical rigor in the foundations of mathematics.[lower-alpha 8] 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 were constructed; 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.[22]

The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, and further explored by Giuseppe Peano; 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. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem.[23]

With all these definitions, it is convenient to include 0 (corresponding to the empty set) as a natural number. Including 0 is now the common convention among set theorists[24] and logicians.[25] Other mathematicians also include 0,[lower-alpha 9] and computer languages often start from zero when enumerating items like loop counters and string- or array-elements.[26][27] On the other hand, many mathematicians have kept the older tradition to take 1 to be the first natural number.[28]

Notation

The set of all natural numbers is standardly denoted N or [3][29] Older texts have occasionally employed J as the symbol for this set.[30]

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:[1][31]

  • Naturals without zero:
  • Naturals with zero:

Alternatively, since the natural numbers naturally form a subset of the integers (often denoted ), they may be referred to as the positive, or the non-negative integers, respectively.[32] To be unambiguous about whether 0 is included or not, sometimes a superscript "" or "+" is added in the former case, and a subscript (or superscript) "0" is added in the latter case:[1]

Properties

This section uses the convention .

Addition

Given the set of natural numbers and the successor function sending each natural number to the next one, one can define addition 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 algebraic structure 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.

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.

Multiplication

Analogously, given that addition has been defined, a multiplication operator can be defined via a × 0 = 0 and a × S(b) = (a × b) + a. This turns 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

Addition and multiplication are compatible, which is expressed in the distribution law: a × (b + c) = (a × b) + (a × c). 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 is not closed under subtraction (that is, subtracting one natural from another does not always result in another natural), means that 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 a + 1 = S(a) and a × 1 = a. Furthermore, has no identity element.

Order

In this section, juxtaposed variables such as ab indicate the product a × b,[33] and the standard order of operations is assumed.

A total order 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 arithmetical operations 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 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 ω (omega).

Division

In this section, juxtaposed variables such as ab indicate the product a × b, 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 a and b with b ≠ 0 there are natural numbers q and r such that

The number q is called the quotient and r is called the remainder 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 (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 a and b, both a + b and a × b are natural numbers.[34]
  • Associativity: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.[35]
  • Commutativity: for all natural numbers a and b, a + b = b + a and a × b = b × a.[36]
  • Existence of identity elements: for every natural number a, a + 0 = a and a × 1 = a.
    • 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
  • Distributivity of multiplication over addition for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c).
  • No nonzero zero divisors: if a and b are natural numbers such that a × b = 0, then a = 0 or b = 0 (or both).
    • If the natural numbers are taken as "excluding 0", and "starting at 1", the "no nonzero zero divisors" property is not satisfied.

Generalizations

Two important generalizations of natural numbers arise from the two uses of counting and ordering: cardinal numbers and ordinal numbers.

  • 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. This concept of "size" relies on maps between sets, such that two sets have the same size, exactly if there exists a bijection between them. The set of natural numbers itself, and any bijective image of it, is said to be countably infinite and to have cardinality aleph-null (ℵ0).
  • Natural numbers are also used as linguistic ordinal numbers: "first", "second", "third", and so forth. This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any well-ordered countably infinite set. 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 order isomorphism (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.

The least ordinal of cardinality ℵ0 (that is, the initial ordinal of ℵ0) is ω but many well-ordered sets with cardinal number ℵ0 have an ordinal number greater than ω.

For finite 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, sequence.

A countable non-standard model of arithmetic satisfying the Peano Arithmetic (that is, the first-order Peano axioms) was developed by Skolem in 1933. The hypernatural numbers are an uncountable model that can be constructed from the ordinary natural numbers via the ultrapower construction.

Georges Reeb used to claim provocatively that "The naïve integers don't fill up" . Other generalizations are discussed in the article on numbers.

Formal definitions

There are two standard methods for formally defining natural numbers. The first one, named for Giuseppe Peano, consists of an autonomous axiomatic theory called Peano arithmetic, based on few axioms called Peano axioms.

The second definition is based on set theory. It defines the natural numbers as specific sets. 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 one to one correspondence between the two sets n and S.

The sets used to define natural numbers satisfy 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 Peano axioms provide a model of Peano arithmetic inside set theory. An important consequence is that, if set theory is consistent (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.

Peano axioms

The five Peano axioms are the following:[37][lower-alpha 10]

  1. 0 is a natural number.
  2. Every natural number has a successor which is also a natural number.
  3. 0 is not the successor of any natural number.
  4. If the successor of equals the successor of , then equals .
  5. 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 is .

Set-theoretic definition

Intuitively, the natural number n is the common property of all sets that have n elements. So, it seems natural to define n as an equivalence class under the relation "can be made in one to one correspondence". Unfortunately, this does not work in 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 n elements that will be called the natural number n.

The following definition was first published by John von Neumann,[38] although Levy attributes the idea to unpublished work of Zermelo in 1916.[39] 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 0 = { }, the empty set.
  • Define the successor S(a) of any set a by S(a) = a ∪ {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.

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

  • 0 = { },
  • 1 = 0 ∪ {0} = {0} = {{ }},
  • 2 = 1 ∪ {1} = {0, 1} = {{ }, {{ }}},
  • 3 = 2 ∪ {2} = {0, 1, 2} = {{ }, {{ }}, {{ }, {{ }}}},
  • n = n−1 ∪ {n−1} = {0, 1, ..., n−1} = {{ }, {{ }}, ..., {{ }, {{ }}, ...}},
  • etc.

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

With this definition, given a natural number n, the sentence "a set S has n elements" can be formally defined as "there exists a bijection from n to S. This formalizes the operation of counting the elements of S. Also, n ≤ m if and only if n is a subset of m. 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 ordinal numbers, including the infinite ones: "each ordinal is the well-ordered set of all smaller ordinals."

If one does not accept the axiom of infinity, 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 Zermelo ordinals.[39] It consists in defining 0 as the empty set, and S(a) = {a}.

With this definition each 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 nth element of a sequence) is immediate. Unlike von Neumann's construction, the Zermelo ordinals do not extend to infinite ordinals.

See also

Number systems
Complex
Real
Rational
Integer
Natural
Zero: 0
One: 1
Prime numbers
Composite numbers
Negative integers
Fraction
Finite decimal
Dyadic (finite binary)
Repeating decimal
Irrational
Algebraic irrational
Transcendental
Imaginary

Notes

  1. ↑ Carothers (2000, p. 3) says, " is the set of natural numbers (positive integers)." Both definitions are acknowledged whenever convenient, and there is no general consensus on whether zero should be included in the natural numbers.[3]
  2. ↑ Any Cauchy sequence in the Reals converges,
  3. ↑ Mendelson (2008, 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."
  4. ↑ Bluman (2010, p. 1): "Numbers make up the foundation of mathematics."
  5. ↑ 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.[11]
  6. ↑ This convention is used, for example, in Euclid's Elements, see D. Joyce's web edition of Book VII.[15]
  7. ↑ 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."[20][21]
  8. ↑ "Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." (Eves 1990, p. 606)
  9. ↑ Mac Lane & Birkhoff (1999, p. 15) include zero in the natural numbers: 'Intuitively, the set of all natural numbers may be described as follows: contains an "initial" number 0; ...'. They follow that with their version of the Peano's axioms.
  10. ↑ Hamilton (1988, pp. 117 ff) calls them "Peano's Postulates" and begins with "1.  0 is a natural number."
    Halmos (1960, 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).
    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)

References

  1. 1 2 3 "Standard number sets and intervals" (PDF). ISO 80000-2:2019. International Organization for Standardization. 19 May 2020. p. 4.
  2. ↑ "natural number". Merriam-Webster.com. Merriam-Webster. Archived from the original on 13 December 2019. Retrieved 4 October 2014.
  3. 1 2 3 Weisstein, Eric W. "Natural Number". mathworld.wolfram.com. Retrieved 11 August 2020.
  4. ↑ Ganssle, Jack G. & Barr, Michael (2003). "integer". Embedded Systems Dictionary. Taylor & Francis. pp. 138 (integer), 247 (signed integer), & 276 (unsigned integer). ISBN 978-1-57820-120-4. Archived from the original on 29 March 2017. Retrieved 28 March 2017 – via Google Books.
  5. ↑ Weisstein, Eric W. "Counting Number". MathWorld.
  6. ↑ "Natural Numbers". Brilliant Math & Science Wiki. Retrieved 11 August 2020.
  7. ↑ "Introduction". Ishango bone. Brussels, Belgium: Royal Belgian Institute of Natural Sciences. Archived from the original on 4 March 2016.
  8. ↑ "Flash presentation". Ishango bone. Brussels, Belgium: Royal Belgian Institute of Natural Sciences. Archived from the original on 27 May 2016.
  9. ↑ "The Ishango Bone, Democratic Republic of the Congo". UNESCO's Portal to the Heritage of Astronomy. Archived from the original on 10 November 2014., on permanent display at the Royal Belgian Institute of Natural Sciences, Brussels, Belgium.
  10. ↑ Ifrah, Georges (2000). The Universal History of Numbers. Wiley. ISBN 0-471-37568-3.
  11. ↑ "A history of Zero". MacTutor History of Mathematics. Archived from the original on 19 January 2013. Retrieved 23 January 2013.
  12. ↑ Mann, Charles C. (2005). 1491: New Revelations of the Americas before Columbus. Knopf. p. 19. ISBN 978-1-4000-4006-3. Archived from the original on 14 May 2015. Retrieved 3 February 2015 – via Google Books.
  13. ↑ Evans, Brian (2014). "Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations". The Development of Mathematics Throughout the Centuries: A brief history in a cultural context. John Wiley & Sons. ISBN 978-1-118-85397-9 – via Google Books.
  14. ↑ Deckers, Michael (25 August 2003). "Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius". Hbar.phys.msu.ru. Archived from the original on 15 January 2019. Retrieved 13 February 2012.
  15. ↑ Euclid. "Book VII, definitions 1 and 2". In Joyce, D. (ed.). Elements. Clark University.
  16. ↑ Mueller, Ian (2006). Philosophy of mathematics and deductive structure in Euclid's Elements. Mineola, New York: Dover Publications. p. 58. ISBN 978-0-486-45300-2. OCLC 69792712.
  17. ↑ Euclid. "Book VII, definition 22". In Joyce, D. (ed.). Elements. Clark University. A perfect number is that which is equal to the sum of its own parts. 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.
  18. ↑ Kline, Morris (1990) [1972]. Mathematical Thought from Ancient to Modern Times. Oxford University Press. ISBN 0-19-506135-7.
  19. ↑ Poincaré, Henri (1905) [1902]. "On the nature of mathematical reasoning". La Science et l'hypothèse [Science and Hypothesis]. Translated by Greenstreet, William John. VI.
  20. ↑ Gray, Jeremy (2008). Plato's Ghost: The modernist transformation of mathematics. Princeton University Press. p. 153. ISBN 978-1-4008-2904-0. Archived from the original on 29 March 2017 – via Google Books.
  21. ↑ Weber, Heinrich L. (1891–1892). "Kronecker". Jahresbericht der Deutschen Mathematiker-Vereinigung [Annual report of the German Mathematicians Association]. pp. 2:5–23. (The quote is on p. 19). Archived from the original on 9 August 2018; "access to Jahresbericht der Deutschen Mathematiker-Vereinigung". Archived from the original on 20 August 2017.
  22. ↑ Eves 1990, Chapter 15
  23. ↑ Kirby, Laurie; Paris, Jeff (1982). "Accessible Independence Results for Peano Arithmetic". Bulletin of the London Mathematical Society. Wiley. 14 (4): 285–293. doi:10.1112/blms/14.4.285. ISSN 0024-6093.
  24. ↑ Bagaria, Joan (2017). Set Theory (Winter 2014 ed.). The Stanford Encyclopedia of Philosophy. Archived from the original on 14 March 2015. Retrieved 13 February 2015.
  25. ↑ Goldrei, Derek (1998). "3". Classic Set Theory: A guided independent study (1. ed., 1. print ed.). Boca Raton, Fla. [u.a.]: Chapman & Hall/CRC. p. 33. ISBN 978-0-412-60610-6.
  26. ↑ Brown, Jim (1978). "In defense of index origin 0". ACM SIGAPL APL Quote Quad. 9 (2): 7. doi:10.1145/586050.586053. S2CID 40187000.
  27. ↑ Hui, Roger. "Is index origin 0 a hindrance?". jsoftware.com. Archived from the original on 20 October 2015. Retrieved 19 January 2015.
  28. ↑ This is common in texts about Real analysis. See, for example, Carothers (2000, p. 3) or Thomson, Bruckner & Bruckner (2008, p. 2).
  29. ↑ "Listing of the Mathematical Notations used in the Mathematical Functions Website: Numbers, variables, and functions". functions.wolfram.com. Retrieved 27 July 2020.
  30. ↑ Rudin, W. (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. p. 25. ISBN 978-0-07-054235-8.
  31. ↑ Grimaldi, Ralph P. (2004). Discrete and Combinatorial Mathematics: An applied introduction (5th ed.). Pearson Addison Wesley. ISBN 978-0-201-72634-3.
  32. ↑ Grimaldi, Ralph P. (2003). A review of discrete and combinatorial mathematics (5th ed.). Boston: Addison-Wesley. p. 133. ISBN 978-0-201-72634-3.
  33. ↑ Weisstein, Eric W. "Multiplication". mathworld.wolfram.com. Retrieved 27 July 2020.
  34. ↑ Fletcher, Harold; Howell, Arnold A. (9 May 2014). Mathematics with Understanding. Elsevier. p. 116. ISBN 978-1-4832-8079-0. ...the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication
  35. ↑ Davisson, Schuyler Colfax (1910). College Algebra. Macmillian Company. p. 2. Addition of natural numbers is associative.
  36. ↑ Brandon, Bertha (M.); Brown, Kenneth E.; Gundlach, Bernard H.; Cooke, Ralph J. (1962). Laidlaw mathematics series. Vol. 8. Laidlaw Bros. p. 25.
  37. ↑ Mints, G.E. (ed.). "Peano axioms". Encyclopedia of Mathematics. Springer, in cooperation with the European Mathematical Society. Archived from the original on 13 October 2014. Retrieved 8 October 2014.
  38. ↑ von Neumann (1923)
  39. 1 2 Levy (1979), p. 52

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