Natural number

Last updated

Natural numbers can be used for counting (one apple, two apples, three apples, ...) Three Baskets.svg
Natural numbers can be used for counting (one apple, two apples, three apples, ...)

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common mathematical terminology, words colloquially used for counting are "cardinal numbers", and words used for ordering are "ordinal numbers". The natural numbers can, at times, appear as a convenient set of codes (labels or "names"), that is, as what linguists call nominal numbers, forgoing many or all of the properties of being a number in a mathematical sense. The set of natural numbers is often denoted by the symbol . [1] [2] [3]

Contents

Some definitions, including the standard ISO 80000-2, [4] [lower-alpha 1] begin the natural numbers with 0, corresponding to the non-negative integers0, 1, 2, 3, ... (sometimes collectively denoted by the symbol , to emphasize that zero is included), whereas others start with 1, corresponding to the positive integers 1, 2, 3, ... (sometimes collectively denoted by the symbol , , or for emphasizing that zero is excluded). [5] [6] [lower-alpha 2]

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). [7] [ dubious ]

The natural numbers are a basis from which many other number sets may be built by extension: the integers, by including (if not yet in) the neutral element 0 and an additive inverse (n) for each nonzero natural number n; the rational numbers, by including a multiplicative inverse (1/n) for each nonzero integer n (and also the product of these inverses by integers); the real numbers by including with the rationals the limits of (converging) Cauchy sequences of rationals; the complex numbers, by including with the real numbers the unresolved square root of minus one (and also the sums and products thereof); and so on. [lower-alpha 3] [lower-alpha 4] These chains of extensions make the natural numbers canonically embedded (identified) 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.

In common language, particularly in primary school education, natural numbers may be called counting numbers [8] 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.

History

Ancient roots

The Ishango bone (on exhibition at the Royal Belgian Institute of Natural Sciences) is believed to have been used 20,000 years ago for natural number arithmetic. Os d'Ishango IRSNB.JPG
The Ishango bone (on exhibition at the Royal Belgian Institute of Natural Sciences) 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 put down a mark for each object. 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. [12]

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. [14] [15] 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. [16]

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). [18]

Independent studies on numbers also occurred at around the same time in India, China, and Mesoamerica. [19]

Modern definitions

In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school[ which? ] of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker, who summarized his belief as "God made the integers, all else is the work of man". [lower-alpha 7]

In opposition to the Naturalists, 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 1] 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 double-struck capital N symbol, often used to denote the set of all natural numbers (see Glossary of mathematical symbols). U+2115.svg
The double-struck capital N symbol, often used to denote the set of all natural numbers (see Glossary of mathematical symbols).

Mathematicians use N or to refer to the set of all natural numbers. [1] [2] [29] Older texts have also occasionally employed J as the symbol for this set. [30]

Since different properties are customarily associated to the tokens 0 and 1 (e.g., neutral elements for addition and multiplications, respectively), it is important to know which version of natural numbers is employed in the case under consideration. This can be done by explanation in prose, by explicitly writing down the set, or by qualifying the generic identifier with a super- or subscript, [4] [31] for example, like this:

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 subscript (or superscript) "0" is added in the former case, and a superscript "*" is added in the latter case: [5] [4]

Properties

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. Then (ℕ, +) 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 (ℕ*, ×) 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.

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 ab 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 ab, then a + cb + c and acbc.

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:

Infinity

The set of natural numbers is an infinite set. By definition, this kind of infinity is called countable infinity. All sets that can be put into a bijective relation to the natural numbers are said to have this kind of infinity. This is also expressed by saying that the cardinal number of the set is aleph-nought (0). [37]

Generalizations

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

The least ordinal of cardinality 0 (that is, the initial ordinal of 0) is ω but many well-ordered sets with cardinal number 0 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

Peano axioms

Many properties of the natural numbers can be derived from the five Peano axioms: [38] [lower-alpha 9]

  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 . Replacing axiom 5 by an axiom schema, one obtains a (weaker) first-order theory called Peano arithmetic .

Constructions based on set theory

Von Neumann ordinals

In the area of mathematics called set theory, a specific construction due to John von Neumann [39] [40] defines the natural numbers as follows:

  • Set 0 = { }, the empty set,
  • Define S(a) = a ∪ {a} for every set a. S(a) is the successor of a, and S is called the successor function.
  • By the axiom of infinity, there exists a set which contains 0 and is closed under the successor function. Such sets are said to be inductive. The intersection of all such inductive sets is defined to be the set of natural numbers. It can be checked that the set of natural numbers satisfies the Peano axioms.
  • It follows that each natural number is equal to the set of all natural numbers less than it:
  • 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.

With this definition, a natural number n is a particular set with n elements, and nm if and only if n is a subset of m. The standard definition, now called definition of von Neumann ordinals, is: "each ordinal is the well-ordered set of all smaller ordinals."

Also, with this definition, different possible interpretations of notations like n (n-tuples versus mappings of n into ) coincide.

Even if one does not accept the axiom of infinity and therefore cannot accept that the set of all natural numbers exists, it is still possible to define any one of these sets.

Zermelo ordinals

Although the standard construction is useful, it is not the only possible construction. Ernst Zermelo's construction goes as follows: [40]

  • Set 0 = { }
  • Define S(a) = {a},
  • It then follows that
  • 0 = { },
  • 1 = {0} = {{ }},
  • 2 = {1} = {{{}}},
  • n = {n−1} = {{{...}}}, etc.
Each natural number is then equal to the set containing just the natural number preceding it. This is the definition of Zermelo ordinals. Unlike von Neumann's construction, the Zermelo ordinals do not account for infinite ordinals.

See also

Notes

  1. 1 2 Mac Lane & Birkhoff (1999, p. 15) include zero in the natural numbers: 'Intuitively, the set ℕ = {0, 1, 2, ...} of all natural numbers may be described as follows: contains an "initial" number 0; ...'. They follow that with their version of the Peano Postulates.
  2. 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 as the natural numbers. [2]
  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." (Preface(px))
  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. [13]
  6. This convention is used, for example, in Euclid's Elements, see D. Joyce's web edition of Book VII. [17]
  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. 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)

Related Research Articles

Cardinal number Generalization of natural numbers

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The transfinite cardinal numbers, often denoted using the Hebrew symbol (aleph) followed by a subscript, describe the sizes of infinite sets.

Empty set Mathematical set containing no elements

In mathematics, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set.

Integer Number in {..., –2, –1, 0, 1, 2, ...}

An integer is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and 2 are not.

Mathematical induction Form of mathematical proof

Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, .. . ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3),. .. . Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder:

Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung and that from each rung we can climb up to the next one.

Group (mathematics) Algebraic structure with one binary operation

In mathematics, a group is a set equipped with an operation that combines any two elements to form a third element while being associative as well as having an identity element and inverse elements. These three conditions, called group axioms, hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The formulation of the axioms is, however, detached from the concrete nature of the group and its operation. This allows one to handle entities of very different mathematical origins in a flexible way, while retaining essential structural aspects of many objects in abstract algebra and beyond. The ubiquity of groups in numerous areas—both within and outside mathematics—makes them a central organizing principle of contemporary mathematics.

Modular arithmetic Computation modulo a fixed integer

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

Multiplication Arithmetical operation

Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction and division. The result of a multiplication operation is called a product.

Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929. The signature of Presburger arithmetic contains only the addition operation and equality, omitting the multiplication operation entirely. The axioms include a schema of induction.

In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.

Recursion Process of repeating items in a self-similar way

Recursion occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances, it is often done in such a way that no infinite loop or infinite chain of references can occur.

Division by zero The result yielded by a real number when divided by zero

In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as where a is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is no number which, when multiplied by 0, gives a, and so division by zero is undefined. Since any number multiplied by zero is zero, the expression is also undefined; when it is the form of a limit, it is an indeterminate form. Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to is contained in Anglo-Irish philosopher George Berkeley’s criticism of infinitesimal calculus in 1734 in The Analyst.

In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by R. M. Robinson in 1950. It is usually denoted Q. Q is almost PA without the axiom schema of mathematical induction. Q is weaker than PA but it has the same language, and both theories are incomplete. Q is important and interesting because it is a finitely axiomatized fragment of PA that is recursively incompletable and essentially undecidable.

In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics.

In mathematical logic, an ω-consistenttheory is a theory that is not only (syntactically) consistent, but also avoids proving certain infinite combinations of sentences that are intuitively contradictory. The name is due to Kurt Gödel, who introduced the concept in the course of proving the incompleteness theorem.

Gentzen's consistency proof is a result of proof theory in mathematical logic, published by Gerhard Gentzen in 1936. It shows that the Peano axioms of first-order arithmetic do not contain a contradiction, as long as a certain other system used in the proof does not contain any contradictions either. This other system, today called "primitive recursive arithmetic with the additional principle of quantifier-free transfinite induction up to the ordinal ε0", is neither weaker nor stronger than the system of Peano axioms. Gentzen argued that it avoids the questionable modes of inference contained in Peano arithmetic and that its consistency is therefore less controversial.

In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of relevance to proof theory still have computable ordinal notations. However, it is not possible to decide effectively whether a given putative ordinal notation is a notation or not ; various more-concrete ways of defining ordinals that definitely have notations are available.

In proof theory, ordinal analysis assigns ordinals to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory.

Real number Number representing a continuous quantity

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as 2. Included within the irrationals are the real transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more. The set of real numbers is denoted using the symbol R or and is sometimes called "the reals".

In mathematical logic, true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers. This is the theory associated with the standard model of the Peano axioms in the language of the first-order Peano axioms. True arithmetic is occasionally called Skolem arithmetic, though this term usually refers to the different theory of natural numbers with multiplication.

Mathematics is a field of study that investigates topics such as number, space, structure, and change.

References

  1. 1 2 "Compendium of Mathematical Symbols". Math Vault. 1 March 2020. Retrieved 11 August 2020.
  2. 1 2 Weisstein, Eric W. "Natural Number". mathworld.wolfram.com. Retrieved 11 August 2020.
  3. "Natural Numbers". Brilliant Math & Science Wiki. Retrieved 11 August 2020.
  4. 1 2 3 "Standard number sets and intervals". ISO 80000-2:2009. International Organization for Standardization. p. 6.
  5. 1 2 "Comprehensive List of Algebra Symbols". Math Vault. 25 March 2020. Retrieved 11 August 2020..
  6. "natural number". Merriam-Webster.com. Merriam-Webster. Archived from the original on 13 December 2019. Retrieved 4 October 2014.
  7. Ganssle, Jack G. & Barr, Michael (2003). "integer". Embedded Systems Dictionary. 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. integer 1. n. Any whole number.
  8. Weisstein, Eric W. "Counting Number". MathWorld .
  9. "Introduction". Ishango bone. Brussels, Belgium: Royal Belgian Institute of Natural Sciences. Archived from the original on 4 March 2016.
  10. "Flash presentation". Ishango bone. Brussels, Belgium: Royal Belgian Institute of Natural Sciences. Archived from the original on 27 May 2016.
  11. "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.
  12. Ifrah, Georges (2000). The Universal History of Numbers. Wiley. ISBN   0-471-37568-3.
  13. "A history of Zero". MacTutor History of Mathematics. Archived from the original on 19 January 2013. Retrieved 23 January 2013.
  14. 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.
  15. 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.
  16. 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.
  17. Euclid. "Book VII, definitions 1 and 2". In Joyce, D. (ed.). Elements . Clark University. Archived from the original on 5 August 2011.
  18. 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.
  19. Kline, Morris (1990) [1972]. Mathematical Thought from Ancient to Modern Times. Oxford University Press. ISBN   0-19-506135-7.
  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 (2000 , 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. 8. Laidlaw Bros. p. 25.
  37. Weisstein, Eric W. "Cardinal Number". MathWorld .
  38. 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.
  39. von Neumann (1923)
  40. 1 2 Levy (1979) , p. 52 attributes the idea to unpublished work of Zermelo in 1916 and several papers by von Neumann the 1920s.

Bibliography