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 language, words used for counting are "cardinal numbers " and words used for ordering are "ordinal numbers ". Some definitions, including the standard ISO 800002 , begin the natural numbers with 0 , corresponding to the NONNEGATIVE INTEGERS 0, 1, 2, 3, …, whereas others start with 1, corresponding to the POSITIVE INTEGERS 1 , 2 , 3 , …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the WHOLE NUMBERS, but in other writings, that term is used instead for the integers (including negative integers). The natural numbers are the 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 of thereof); and so on. 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, for example in primary school , natural numbers may be called COUNTING NUMBERS both to intuitively exclude the negative integers and zero, and also to contrast the discreteness of counting to the continuity of measurement , established by the real numbers . The natural numbers can, at times, appear as a convenient set of names (labels), that is, as what linguists call nominal numbers , foregoing many or all of the properties of being a number in a mathematical sense. CONTENTS * 1 History * 1.1 Ancient roots * 1.2 Modern definitions * 2 Notation * 3 Properties * 3.1 Addition * 3.2 Multiplication * 3.3 Relationship between addition and multiplication * 3.4 Order * 3.5 Division * 3.6 Algebraic properties satisfied by the natural numbers * 4 Generalizations * 5 Formal definitions * 5.1 Peano axioms * 5.2 Constructions based on set theory * 5.2.1 Von Neumann construction * 5.2.2 Other constructions * 6 See also * 7 Notes * 8 References * 9 External links 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. 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 the powers of 10
up to over 1 million. A stone carving from
Karnak
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
placevalue notation (within other numbers) dates back as early as 700
BC by the Babylonians, but they omitted such a digit when it would
have been the last symbol in the number. The
Olmec
The first systematic study of numbers as abstractions is usually
credited to the Greek philosophers
Pythagoras
Independent studies also occurred at around the same time in
India
MODERN DEFINITIONS This section NEEDS ADDITIONAL CITATIONS FOR VERIFICATION . Please help improve this article by adding citations to reliable sources . Unsourced material may be challenged and removed. (October 2014) (Learn how and when to remove this template message ) In
19th century
Europe
In opposition to the Naturalists, the constructivists saw a need to
improve the logical rigor in the foundations of mathematics . In the
1860s,
Hermann Grassmann
Settheoretical definitions of natural numbers were initiated by Frege and he initially defined a natural number as the class of all sets that are in onetoone correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell\'s paradox . Therefore, this formalism was modified so that a natural number is defined as a particular set, and any set that can be put into onetoone correspondence with that set is said to have that number of elements. The second class of definitions was introduced by
Giuseppe Peano
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 and logicians . Other mathematicians also include 0 although many have kept the older tradition and take 1 to be the first natural number. Computer scientists often start from zero when enumerating items like loop counters and string or array elements. NOTATION The doublestruck capital N symbol, often used to denote the set of all natural numbers (see List of mathematical symbols ). Mathematicians use N or ℕ (an N in blackboard bold ) to refer to the set of all natural numbers. Older texts have also occasionally employed J as the symbol for this set. This set is countably infinite : it is infinite but countable by definition. This is also expressed by saying that the cardinal number of the set is alephnaught (ℵ0). To be unambiguous about whether 0 is included or not, sometimes an index (or superscript) "0" is added in the former case, and a superscript "*" or subscript ">0" is added in the latter case: ℕ0 = ℕ0 = {0, 1, 2, …} ℕ* = ℕ+ = ℕ1 = ℕ>0 = {1, 2, …}. Alternatively, natural numbers may be distinguished from positive integers with the index notation, but it must be understood by context that since both symbols are used, the natural numbers contain zero. ℕ = {0, 1, 2, …}. ℤ+= {1, 2, …}. PROPERTIES ADDITION One can recursively define an addition operator on the natural numbers by setting a + 0 = a and a + S(b) = S(a + b) for all a, b. Here S should be read as "successor ". This turns the natural numbers (ℕ, +) into a commutative monoid with identity element 0, the socalled free object with one generator. This monoid satisfies the cancellation property and can be embedded in a group (in the mathematical sense of the word 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 ℕ is not closed under subtraction (i.e., subtracting one natural from another does not always result in another natural), means that 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 a + 1 = S(a) and a × 1 = a. ORDER In this section, juxtaposed variables such as ab indicate the product a × b, 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 wellordered : every nonempty set of natural numbers has a least element. The rank among wellordered 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 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 a = bq + r and r < b. The number q is called the quotient and r is called the remainder of division of a by b. The numbers q and r are uniquely determined by a and b. This Euclidean division is key to 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.
*
Associativity : for all natural numbers a, b, and c, a + (b + c) =
(a + b) + c and a × (b × c) = (a × b) × c.
*
Commutativity
GENERALIZATIONS Two generalizations of natural numbers arise from the two uses: * A natural number can be used to express the size of a finite set; more generally a cardinal number is a measure for the size of a set also suitable for infinite sets; this refers to a concept of "size" such that if there is a bijection between two sets they have the same size . The set of natural numbers itself and any other countably infinite set has cardinality alephnull (ℵ0). * Linguistic ordinal numbers "first", "second", "third" can be assigned to the elements of a totally ordered finite set, and also to the elements of wellordered countably infinite sets like the set of natural numbers itself. This can be generalized to ordinal numbers which describe the position of an element in a wellordered set in general. An ordinal number is also used to describe the "size" of a wellordered set, in a sense different from cardinality: if there is an order isomorphism between two wellordered 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. Many wellordered sets with cardinal number ℵ0 have an ordinal number greater than ω (the latter is the lowest possible). The least ordinal of cardinality ℵ0 (i.e., the initial ordinal ) is ω. For finite wellordered sets, there is onetoone 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 nonstandard model of arithmetic satisfying the Peano Arithmetic (i.e., the firstorder 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 ℕ. Other generalizations are discussed in the article on numbers . FORMAL DEFINITIONS PEANO AXIOMS Main article: Peano axioms Many properties of the natural numbers can be derived from the Peano axioms . * Axiom One: 0 is a natural number. * Axiom Two: Every natural number has a successor. * Axiom Three: 0 is not the successor of any natural number. * Axiom Four: If the successor of x equals the successor of y, then x equals y. * Axiom Five (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. Replacing Axiom Five by an axiom schema one obtains a (weaker) firstorder theory called Peano Arithmetic. CONSTRUCTIONS BASED ON SET THEORY Main article: Settheoretic definition of natural numbers Von Neumann Construction In the area of mathematics called set theory , a special case of the von Neumann ordinal construction 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 n ≤ m if and only if n is a subset of m. Also, with this definition, different possible interpretations of notations like ℝn (ntuples versus mappings of n into ℝ) 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. Other Constructions Although the standard construction is useful, it is not the only possible construction. Zermelo 's construction goes as follows: * 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. SEE ALSO *
Mathematics
*
Integer
NOTES * ^ A B "Standard number sets and intervals". ISO 800002:2009.
International Organization for Standardization
* ^ The English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891/92, 19, quoting from a lecture of Kronecker's of 1886." Gray, Jeremy (2008), Plato\'s Ghost: The Modernist Transformation of Mathematics, Princeton University Press, p. 153 Weber, Heinrich L. 18912. Kronecker. Jahresbericht der Deutschen MathematikerVereinigung 2:523. (The quote is on p. 19.) * ^ "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) * ^ Eves 1990 , Chapter 15 * ^ L. Kirby; J. Paris, Accessible Independence Results for Peano Arithmetic, Bulletin of the London Mathematical Society 14 (4): 285. doi:10.1112/blms/14.4.285, 1982. * ^ Bagaria, Joan. "Set Theory". The Stanford Encyclopedia of Philosophy (Winter 2014 Edition). * ^ Goldrei, Derek (1998). "3". Classic set theory : a guided independent study (1. ed., 1. print ed.). Boca Raton, Fla. : Chapman & Hall/CRC. p. 33. ISBN 0412606100 . * ^ This is common in texts about Real analysis . See, for example, Carothers (2000 , p. 3) or Thomson, Bruckner & Bruckner (2000 , p. 2). * ^ Brown, Jim (1978). "In Defense of Index Origin 0". ACM SIGAPL APL Quote Quad. 9 (2): 7. doi :10.1145/586050.586053 . Retrieved 19 January 2015. * ^ Hui, Roger. "Is Index Origin 0 a Hindrance?". jsoftware.com. Retrieved 19 January 2015. * ^ Rudin, W. (1976). Principles of Mathematical Analysis (PDF). New York: McGrawHill. p. 25. ISBN 9780070542358 . * ^ Weisstein, Eric W. "Cardinal Number". MathWorld . * ^ Grimaldi, Ralph P. (2003). A review of discrete and combinatorial mathematics (5th ed.). Boston, MA: AddisonWesley. p. 133. ISBN 9780201726343 . * ^ G.E. Mints (originator), "Peano axioms", Encyclopedia of Mathematics, Springer , in cooperation with the European Mathematical Society , retrieved 8 October 2014 * ^ Hamilton (1988) calls them "Peano's Postulates" and begins with "1. 0 is a natural number." (p. 117f) Halmos (1960) 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). (p. 46) Morash (1991) gives "a twopart axiom" in which the natural numbers begin with 1. (Section 10.1: An Axiomatization for the System of Positive Integers) * ^ Von Neumann 1923 REFERENCES * Bluman, Allan (2010), PreAlgebra DeMYSTiFieD (Second ed.),
McGrawHill Professional
* Carothers, N.L. (2000), Real analysis, Cambridge University Press,
ISBN 0521497566
* Clapham, Christopher; Nicholson, James (2014), The Concise Oxford
Dictionary of
Mathematics
* Dedekind, Richard (1963), Essays on the Theory of Numbers, Dover, ISBN 0486210103 * Dedekind, Richard (2007), Essays on the Theory of Numbers, Kessinger Publishing, LLC, ISBN 054808985X * Eves, Howard (1990), An Introduction to the History of
Mathematics
* Von Neumann, Johann (1923), "Zur Einführung der transfiniten Zahlen", Acta litterarum ac scientiarum Ragiae Universitatis Hungaricae FranciscoJosephinae, Sectio scientiarum mathematicarum, 1: 199–208 * Von Neumann, John (January 2002) , "On the introduction of transfinite numbers", in Jean van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic, 18791931 (3rd ed.), Harvard University Press, pp. 346–354, ISBN 0674324498  English translation of von Neumann 1923 . EXTERNAL LINKS * Hazewinkel, Michiel, ed. (2001), "Natural number", Encyclopedia of
Mathematics
