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In mathematics, a natural number (also called counting number) can mean either an element of the set {1, 2, 3, . Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity In mathematics Two has many properties in Mathematics. An Integer is called Even if it is divisible by 2 ---- In mathematics Three is the first odd Prime number, and the second smallest prime . . } (the positive integers) or an element of the set {0, 1, 2, 3, . A negative number is a Number that is less than zero, such as −2 The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French . . } (the non-negative integers). A negative number is a Number that is less than zero, such as −2 The former is generally used in number theory, while the latter is preferred in mathematical logic, set theory, and computer science. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their A more formal definition will follow.

Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), and they can be used for ordering ("this is the 3rd largest city in the country"). Counting is the mathematical action of repeatedly adding (or subtracting one usually to find out how many objects there are or to set aside a desired number of objects (starting In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement

Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Problems concerning counting, such as Ramsey theory, are studied in combinatorics. This article provides an introduction For a more detailed and technical article see Ramsey's theorem. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects

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

Contents

History of natural numbers and the status of zero

The natural numbers had their origins in the words used to count things, beginning with the number one.

The first major advance in abstraction was the use of numerals to represent numbers. A numeral system (or system of numeration) is a Mathematical notation for representing numbers of a given set by symbols in a consistent manner This allowed systems to be developed for recording large numbers. For example, the Babylonians developed a powerful place-value system based essentially on the numerals for 1 and 10. Babylonia was an Amorite state in lower Mesopotamia (modern southern Iraq) with Babylon as its capital A positional notation or place-value notation system is a Numeral system in which each position is related to the next by a Constant multiplier a The ancient Egyptians had a system of numerals with distinct hieroglyphs for 1, 10, and all the powers of 10 up to one million. The History of Ancient Egypt spans the period from the early predynastic settlements of the northern Nile Valley to the Roman conquest in 30 Egyptian hieroglyphs (ˈhaɪərəʊɡlɪf from Greek grc-Grek ἱερογλύφος " sacred carving " also hieroglyphic = grc-Grek A stone carving from Karnak, dating from around 1500 BC 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 Karnak temple complex, universally known only as Karnak, describes a vast conglomeration of ruined temples chapels pylons and other buildings The Louvre Museum (Musée du Louvre located in Paris is the world's most visited art museum a historic monument and a national museum of France

A much later advance in abstraction was the development of the idea of zero as a number with its own numeral. A zero digit had been used in place-value notation as early as 700 BC by the Babylonians, but, they omitted it when it would have been the last symbol in the number. In Mathematics and Computer science, a digit is a symbol (a number symbol e [1] The Olmec and Maya civilization used zero as a separate number as early as 1st century BC, apparently developed independently, but this usage did not spread beyond Mesoamerica. The Olmec were an ancient Pre-Columbian people living in the Tropical lowlands of south-central Mexico, in what are roughly the modern-day states The Maya civilization is a Mesoamerican Civilization, noted for the only known fully developed written language of the Pre-Columbian Americas Mesoamerica or Meso-America (Mesoamérica is a Region extending approximately from central Mexico to Honduras and Nicaragua, defined The concept as used in modern times originated with the Indian mathematician Brahmagupta in 628. India, officially the Republic of India (भारत गणराज्य inc-Latn Bhārat Gaṇarājya; see also other Indian languages) is a country Brahmagupta ( (598–668 was an Indian mathematician and astronomer. Nevertheless, medieval computists (calculators of Easter), beginning with Dionysius Exiguus in 525, used zero as a number without using a Roman numeral to write it. Computus ( Latin for Computation) is the Calculation of the date of Easter in the Christian calendar. Easter ( Greek: Πάσχα Pascha or Pasxa) is the most important religious feast in the Christian Liturgical year. Dionysius Exiguus ( Dennis the Little or Dennis the Short, meaning humble (c Roman numerals are a Numeral system originating in ancient Rome, adapted from Etruscan numerals. Instead nullus, the Latin word for "nothing", was employed. The first systematic study of numbers as abstractions (that is, as abstract entities) is usually credited to the Greek philosophers Pythagoras and Archimedes. --> Abstraction is the process or result of generalization by reducing the information An entity is something that has a distinct separate Existence, though it need not be a material existence The term ancient Greece refers to the period of Greek history lasting from the Greek Dark Ages ca "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor. Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer However, independent studies also occurred at around the same time in India, China, and Mesoamerica. India, officially the Republic of India (भारत गणराज्य inc-Latn Bhārat Gaṇarājya; see also other Indian languages) is a country China ( Wade-Giles ( Mandarin) Chung¹kuo² is a cultural region, an ancient Civilization, and depending on perspective a National Mesoamerica or Meso-America (Mesoamérica is a Region extending approximately from central Mexico to Honduras and Nicaragua, defined

In the nineteenth century, a set-theoretical definition of natural numbers was developed. A definition is a statement of the meaning of a Word or Phrase. With this definition, it was convenient to include zero (corresponding to the empty set) as a natural number. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members Including zero in the natural numbers is now the common convention among set theorists, logicians and computer scientists. Logic is the study of the principles of valid demonstration and Inference. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Other mathematicians, such as number theorists, have kept the older tradition and take 1 to be the first natural number. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes

Notation

Mathematicians use N or \mathbb{N} (an N in blackboard bold, displayed as ℕ in Unicode) to refer to the set of all natural numbers. Blackboard bold is a Typeface style often used for certain symbols in Mathematics and Physics texts in which certain lines of the symbol (usually vertical This set is countably infinite: it is infinite but countable by definition. In Set theory, an infinite set is a set that is not a Finite set. This is also expressed by saying that the cardinal number of the set is aleph-nullhttp://en.wikipedia.org../../../../articles/a/l/e/Aleph_number.html#Aleph-null (\aleph_0). This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English.

To be unambiguous about whether zero is included or not, sometimes an index "0" is added in the former case, and a superscript "*" is added in the latter case:

\mathbb{N}0 = { 0, 1, 2, . . . } ; \mathbb{N}* = { 1, 2, . . . }.

(Sometimes, an index or superscript "+" is added to signify "positive". This article is about the terms 'subscript' and 'superscript' as used in typography However, this is often used for "nonnegative" in other cases, as ℝ+ = [0,∞) and ℤ+ = { 0, 1, 2,. . . }, at least in European literature. The notation "*", however, is standard for nonzero or rather invertible elements. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to )

Some authors who exclude zero from the naturals use the term whole numbers, denoted \mathbb{W}, for the set of nonnegative integers. Others use the notation \mathbb{P} for the positive integers.

Set theorists often denote the set of all natural numbers by a lower-case Greek letter omega: ω. OMEGA is the premier Counter-terrorism unit of Latvia. Founded in 1992 OMEGA cooperates with many other counter-terrorism units over the world This stems from the identification of an ordinal number with the set of ordinals that are smaller. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. When this notation is used, zero is explicitly included as a natural number.

Algebraic properties

addition multiplication
closure: a + b   is a natural number a × b   is a natural number
associativity: a + (b + c)  =  (a + b) + c a × (b × c)  =  (a × b) × c
commutativity: a + b  =  b + a a × b  =  b × a
existence of an identity element: a + 0  =  a a × 1  =  a
distributivity: a × (b + c)  =  (a × b) + (a × c)
No zero divisors: if ab = 0, then either a = 0 or b = 0 (or both)

Formal definitions

Main article: Set-theoretic definition of natural numbers

Historically, the precise mathematical definition of the natural numbers developed with some difficulty. In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set In Mathematics, associativity is a property that a Binary operation can have In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0 Several ways have been proposed to define the Natural numbers using Set theory. The Peano postulates state conditions that any successful definition must satisfy. In Mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of Axioms for the Natural Certain constructions show that, given set theory, models of the Peano postulates must exist. In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models

Peano axioms

It should be noted that the "0" in the above definition need not correspond to what we normally consider to be the number zero. "0" simply means some object that when combined with an appropriate successor function, satisfies the Peano axioms. All systems that satisfy these axioms are isomorphic, the name "0" is used here for the first element, which is the only element that is not a successor. For example, the natural numbers starting with one also satisfy the axioms.

Constructions based on set theory

A standard construction

A standard construction in set theory, a special case of the von Neumann ordinal construction, is to define the natural numbers as follows:

We set 0 := { }, the empty set,
and define S(a) = a ∪ {a} for every set a. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members S(a) is the successor of a, and S is called the successor function.
If the axiom of infinity holds, then the set of all natural numbers exists and is the intersection of all sets containing 0 which are closed under this successor function. In Axiomatic set theory and the branches of Logic, Mathematics, and Computer science that use it the axiom of infinity is one of the Axioms
If the set of all natural numbers exists, then it satisfies the Peano axioms. In Mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of Axioms for the Natural
Each natural number is then equal to the set of natural numbers less than it, so that
  • 0 = { }
  • 1 = {0} = {{ }}
  • 2 = {0,1} = {0, {0}} = {{ }, {{ }}}
  • 3 = {0,1,2} = {0, {0}, {0, {0}}} = {{ }, {{ }}, {{ }, {{ }}}}
  • n = {0,1,2,. . . ,n−2,n−1} = {0,1,2,. . . ,n−2} ∪ {n−1} = (n−1) ∪ {n−1}
and so on. When a natural number is used as a set, this is typically what is meant. Under this definition, there are exactly n elements (in the naïve sense) in the set n and n ≤ m (in the naïve sense) if and only if n is a subset of m. ↔
Also, with this definition, different possible interpretations of notations like Rn (n-tuples versus mappings of n into R) coincide.
Even if the axiom of infinity fails and the set of all natural numbers does not exist, it is possible to define what it means to be one of these sets. A set n is a natural number means that it is either 0 (empty) or a successor, and each of its elements is either 0 or the successor of another of its elements.

Other constructions

Although the standard construction is useful, it is not the only possible construction. For example:

one could define 0 = { }
and S(a) = {a},
producing
0 = { }
1 = {0} = {{ }}
2 = {1} = {{{ }}}, etc.

Or we could even define 0 = {{ }}

and S(a) = a U {a}
producing
0 = {{ }}
1 = {{ }, 0} = {{ }, {{ }}}
2 = {{ }, 0, 1}, etc.

Arguably the oldest set-theoretic definition of the natural numbers is the definition commonly ascribed to Frege and Russell under which each concrete natural number n is defined as the set of all sets with n elements. Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin  &ndash 26 July 1925 Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian This may appear circular, but can be made rigorous with care. Define 0 as {{}} (clearly the set of all sets with 0 elements) and define σ(A) (for any set A) as \{x \cup \{y\} \mid x \in A \wedge y \not\in x\}. Then 0 will be the set of all sets with 0 elements, 1 = σ(0) will be the set of all sets with 1 element, 2 = σ(1) will be the set of all sets with 2 elements, and so forth. The set of all natural numbers can be defined as the intersection of all sets containing 0 as an element and closed under σ (that is, if the set contains an element n, it also contains σ(n)). This definition does not work in the usual systems of axiomatic set theory because the collections involved are too large (it will not work in any set theory with the axiom of separation); but it does work in New Foundations (and in related systems known to be consistent) and in some systems of type theory. In Axiomatic set theory and the branches of Logic, Mathematics, and Computer science that use it the axiom schema of specification, axiom In Mathematical logic, New Foundations ( NF) is an Axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of In Mathematics, Logic and Computer science, type theory is any of several Formal systems that can serve as alternatives to Naive set theory

For the rest of this article, we follow the standard construction described above.

Properties

One can recursively define an addition on the natural numbers by setting a + 0 = a and a + S(b) = S(a + b) for all a, b. Addition of natural numbers is the most basic arithmetic binary operation This turns the natural numbers (N, +) into a commutative monoid with identity element 0, the so-called free monoid with one generator. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In Mathematics, the idea of a free object is one of the basic concepts of Abstract algebra. This monoid satisfies the cancellation property and can be embedded in a group. In Mathematics, the notion of cancellative is a generalization of the notion of Invertible. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element The smallest group containing the natural numbers is the integers. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

If we define 1 := S(0), then b + 1 = b + S(0) = S(b + 0) = S(b). That is, b + 1 is simply the successor of b.

Analogously, given that addition has been defined, a multiplication × 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. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 Addition and multiplication are compatible, which is expressed in the distribution law: a × (b + c) = (a × b) + (a × c). In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Abstract algebra, a semiring is an Algebraic structure similar to a ring, but without the requirement that each element must have an Additive inverse Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative.

If we interpret the natural numbers as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that we start with a + 1 = S(a) and a × 1 = a.

For the remainder of the article, we write ab to indicate the product a × b, and we also assume the standard order of operations. In Algebra and Computer programming, when a number or expression is both preceded and followed by a Binary operation, a rule is required for which operation

Furthermore, one defines a total order on the natural numbers by writing a ≤ b if and only if there exists another natural number c with a + c = b. In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation In Mathematics, an inequality is a statement about the relative size or order of two objects or about whether they are the same or not (See also equality 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. In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every The rank among well-ordered sets is expressed by an ordinal number; for the natural numbers this is expressed as "ω". In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set.

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 we can find natural numbers q and r such that

a = bq + r and r < b. In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication.

The number q is called the quotient and r is called the remainder of division of a by b. In Mathematics, a quotient is the result of a division. For example when dividing 6 by 3 the quotient is 2 while 6 is called the dividend, and 3 the In Arithmetic, when the result of the division of two Integers cannot be expressed with an integer Quotient, the remainder is the amount "left The numbers q and r are uniquely determined by a and b. This, the Division algorithm, is key to several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory. The division algorithm is a Theorem in Mathematics which precisely expresses the outcome of the usual process of division of Integers The name In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without In Number theory, the Euclidean algorithm (also called Euclid's algorithm) is an Algorithm to determine the Greatest common divisor (GCD

The natural numbers including zero form a commutative monoid under addition (with identity element zero), and under multiplication (with identity element one). In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that

Generalizations

Two generalizations of natural numbers arise from the two uses:

\aleph_0 and ω have to be distinguished because many well-ordered sets with cardinal number \aleph_0 have a higher ordinal number than ω, for example, \omega^{\omega^{\omega6+42}\cdot1729+\omega^9+88}\cdot3+\omega^{\omega^\omega}\cdot5+65537; ω is the lowest possible value (the initial ordinal). The Von Neumann cardinal assignment is a Cardinal assignment which uses Ordinal numbers For a Well-ordered set U, we define its

For finite well-ordered sets there is one-to-one correspondence between ordinal and cardinal number; therefore they can both be expressed by the same natural number, the number of elements of the set. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence. In Mathematics, a sequence is an ordered list of objects (or events

Other generalizations are discussed in the article on numbers. A number is an Abstract object, tokens of which are Symbols used in Counting and measuring.

References

  1. ^ ". . .  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. [1]"

External links

Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important Project Gutenberg, abbreviated as PG, is a volunteer effort to Digitize, archive and distribute Cultural works

Dictionary

natural number

-noun

  1. (mathematics) a member of the set of natural numbers - a positive integer (set of [1,2,3,...])
  2. (mathematics) a member of the set of natural numbers - a non-negative integer (set of [0,1,2,3,...]).
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