# Encyclopedia

.
Natural numbers can be used for counting (one apple, two apples, three apples, ...
^ NaturalNumber ONE The natural number one.
• NaturalNumber (JML and MultiJava documentation) 1 February 2010 3:28 UTC www.eecs.ucf.edu [Source type: Academic]

^ One apple, two apples, three apples.
• Ratio of natural numbers. Evolution of the real numbers. 1 February 2010 3:28 UTC www.themathpage.com [Source type: Original source]

^ I said, "The cardinal numbers are used for counting.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

) from top to bottom.
.In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {1, 2, 3, ...^ Hence there exist two natural numbered u and v such that a + u = b and c + v = d.
• Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]

^ According to this convention, zero is not a natural number.
• Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]

^ What are the natural numbers or positive integers?
• Goalfinder - Integers, Natural and Whole Numbers ( Math Software ) - Animated Easy Science, Technology Software, Online Education, medical, K12 animation, & e-Learning 1 February 2010 3:28 UTC www.goalfinder.com [Source type: Academic]

} according to the traditional definition or the set of .non-negative integers {0, 1, 2, ...^ Hence the word “count.” FACT: Whole Numbers are non-negative integers that are uncut, undivided, and not in pieces (0, 1, 2, 3…).
• Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]

^ Stepan Wolfram recommends using the terms "positive integers" and "non-negative integers" to avoid confusion.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ This is not tautological, unlike expressions like ‘Whole Numbers are non-negative integers that are uncut, undivided, and not in pieces (0, 1, 2, 3…).
• Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]

} according to a definition first appearing in the nineteenth century.
.Natural numbers have two main purposes: counting ("there are 6 coins on the table") and ordering ("this is the 3rd largest city in the country").^ Hence there exist two natural numbered u and v such that a + u = b and c + v = d.
• Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]

^ Hence, the natural numbers are totally ordered.
• Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]

^ Consider ordered pairs of natural numbers (a, b).
• Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]

.These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively.^ We are explaining how the ordinal numbers -- third, fourth, fifth, and so on -- name the parts of the cardinal numbers.
• Ratio of natural numbers. Evolution of the real numbers. 1 February 2010 3:28 UTC www.themathpage.com [Source type: Original source]

^ The natural numbers have two forms, cardinal and ordinal .
• Ratio of natural numbers. Evolution of the real numbers. 1 February 2010 3:28 UTC www.themathpage.com [Source type: Original source]

^ Some of these tests are not perfect: there may be some composite numbers, called pseudoprimes for the respective test, that will be declared "probably prime" no matter what witness is chosen.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

(See English numerals.) .A more recent notion is that of a nominal number, which is used only for naming.^ As you say, while it is “possible” to have a negative base, it’s likely not very useful because it only works for a small set of numbers.
• Demystifying the Natural Logarithm (ln) | BetterExplained 1 February 2010 3:28 UTC betterexplained.com [Source type: General]

^ Note that in the above-quoted treatment, natural numbers were used not only for illustration as data items but also in the construct ‘a n ’ as exponents.
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

^ If the assertion “There is only one zero” is taken as something else (something more general) than “In any system of X-numbers, the X-number zero is unique”, then it is false .
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

.Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory.^ The study of prime numbers is part of number theory, the branch of mathematics which encompasses the study of natural numbers.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ The distribution of the prime numbers .
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ Whatever we do with natural numbers ultimately relates to properties of exponentiation of operations.
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

Problems concerning counting and ordering, such as partition enumeration, are studied in combinatorics.

## 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 1.[1]
.The first major advance in abstraction was the use of numerals to represent numbers.^ The first major advance in abstraction was the use of numerals to represent numbers.
• Natural number - Psychology Wiki 1 February 2010 3:28 UTC psychology.wikia.com [Source type: Reference]
• WikiSlice 1 February 2010 3:28 UTC dev.laptop.org [Source type: Reference]

^ The first systematic study of numbers as abstractions (that is, as abstract entities) is usually credited to the Greek philosophers Pythagoras and Archimedes .
• WikiSlice 1 February 2010 3:28 UTC dev.laptop.org [Source type: Reference]

^ Nevertheless, zero was used as a number by all medieval computists (calculators of Easter) beginning with Dionysius Exiguus in 525, but in general no Roman numeral was used to write it.
• WikiSlice 1 February 2010 3:28 UTC dev.laptop.org [Source type: Reference]

.This allowed systems to be developed for recording large numbers.^ This allowed systems to be developed for recording large numbers.
• Natural number - Psychology Wiki 1 February 2010 3:28 UTC psychology.wikia.com [Source type: Reference]
• WikiSlice 1 February 2010 3:28 UTC dev.laptop.org [Source type: Reference]

^ In Spain, a large number of photovoltaic systems were damaged.
• The FINANCIAL - Few major natural catastrophe losses in 2009 General trend confirmed by large number of weather extr 1 February 2010 3:28 UTC finchannel.com [Source type: News]
• Munich Re - Few major natural catastrophe losses in 2009 General trend confirmed by large number of weather extremes 1 February 2010 3:28 UTC www.munichre.com [Source type: FILTERED WITH BAYES]

^ Over time, several systems for counting things were developed; the first of which was the natural numbers .
• Arithmetic/Introduction to Natural Numbers - Wikibooks, collection of open-content textbooks 1 February 2010 3:28 UTC en.wikibooks.org [Source type: FILTERED WITH BAYES]

.The ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, and all the powers of 10 up to one million.^ MW - Megawatt - A megawatt is a unit of electrical power equal to one million watts.
• Natural Gas Marketwatch Glossary 1 February 2010 3:28 UTC www.spragueenergy.com [Source type: FILTERED WITH BAYES]

^ The Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to a specified integer.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ Note also the difference between the even powers and the odd powers: the numerators are negated for the even powers as compared to the Harmonic Sum which is an odd power of one .
• Sums of Powers of the Natural Numbers 1 February 2010 3:28 UTC members.bex.net [Source type: Reference]

The Babylonians had a place-value system based essentially on the numerals for 1 and 10. 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.
.A much later advance in abstraction was the development of the idea of zero as a number with its own numeral.^ In any case, Wittgenstein did not explicitly develop further his idea of the system of natural numbers, although this system played an important rôle in his metamathematical investigations.
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

^ For any natural number n other than zero, n, The exception in the wording reminds me; why isn't 0 defined to be its own additive inverse and 1 its own multiplicative inverse, for simplicity?
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

.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.^ We will use B n to notate the n'th Bernoulli Number .
• Sums of Powers of the Natural Numbers 1 February 2010 3:28 UTC members.bex.net [Source type: Reference]

^ It is possible to quickly check whether a given large number (say, up to a few thousand digits) is prime using probabilistic primality tests.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ When creating a field that would be used for small numbers, you can set its data type to either DB_BYTE or dbByte .
• Lesson 16: Data Entry and Natural Numbers 1 February 2010 3:28 UTC www.functionx.com [Source type: Reference]

[2] .The Olmec and Maya civilization used zero developed independently as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica.^ Whenever we needed 0, we would use the Notation of N with a small zero to signify that we assumed 0 to be present in the natural numbers.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ In any case, Wittgenstein did not explicitly develop further his idea of the system of natural numbers, although this system played an important rôle in his metamathematical investigations.
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

^ At the start of the 19th century, Legendre and Gauss independently conjectured that as x tends to infinity, the number of primes up to x is asymptotic to x/log(x), where log(x) is the natural logarithm of x.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

.The concept as used in modern times originated with the Indian mathematician Brahmagupta in 628. Nevertheless, medieval computers (e.g.^ We frequently use concepts like “the n th derivative of a function”, “the n th iterated kernel of an integral equation”, “recurrence relations”, “applying a formula n times”, etc.
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

people who calculated the date of .Easter), beginning with Dionysius Exiguus in 525, used zero as a number without using a Roman numeral to write it.^ Whenever we needed 0, we would use the Notation of N with a small zero to signify that we assumed 0 to be present in the natural numbers.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ They are numbered with Roman numerals, beginning in the upper right quadrant and proceeding in a counter-clockwise rotation.
• Math Glossary N-Z 1 February 2010 3:28 UTC www.blc.edu [Source type: Reference]

^ You write: "The integers (written Z) are all of the numbers, both larger and smaller than zero, that can be written without fractions."
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

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.^ After the Greeks, little happened with the study of prime numbers until the 17th century.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ ObsessiveMathsFreak wrote: "For this reason, the Greeks never really studied true 'numbers' and their properties.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

.Note that many Greek mathematicians did not consider 1 to be "a number", so to them 2 was the smallest number.^ Many mathematicians have worked on primality tests for large numbers, often restricted to specific number forms.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ The oldest known proof for the statement that there are infinitely many prime numbers is given by the Greek mathematician Euclid in his Elements (Book IX, Proposition 20).
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ If you consider as they did, that a real number means a length of a line segment, then you can do things with numbers that don't make much sense with "lengths".
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

[3]
.Independent studies also occurred at around the same time in India, China, and Mesoamerica.^ The same prime factor may occur multiple times.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

.Several set-theoretical definitions of natural numbers were developed in the 19th century.^ Take the set of natural numbers N .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ We usually write N for the set of natural numbers.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ I, too, had been under the impression that zero was not a natural, or counting, number but, rather, the first in the set of whole numbers.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

.With these definitions it was convenient to include 0 (corresponding to the empty set) as a natural number.^ Take the set of natural numbers N .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ These labels are what we recognise as Natural Numbers.
• Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]

^ We usually write N for the set of natural numbers.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

.Including 0 is now the common convention among set theorists, logicians, and computer scientists.^ So the set theorists and the computer scientists should just conform… .
• Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]

^ A computer science definition or a set theorist one?
• Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]

.Many other mathematicians also include 0, although some have kept the older tradition and take 1 to be the first natural number[4].^ Take the set of natural numbers N .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ Many numbers occur in nature, and inevitably some of these are prime.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ There are a number of different kinds of natural law legal theories, differing from each other with respect to the role that morality plays in determining the authority of legal norms.
• Natural Law [Internet Encyclopedia of Philosophy] 1 February 2010 3:28 UTC www.utm.edu [Source type: Original source]

.Sometimes the set of natural numbers with 0 included is called the set of whole numbers or counting numbers.^ Location: Natural History -- Call Number: 581.529 R313 .

^ Besides the Byte and the integer, another natural number supported in the libraries is called Long or Long Integer .
• Lesson 16: Data Entry and Natural Numbers 1 February 2010 3:28 UTC www.functionx.com [Source type: Reference]

^ Location: Natural History -- Call Number: 639.92 G375h .

## Notation

.Mathematicians use N or $\mathbb{N}$ (an N in blackboard bold, displayed as in Unicode) to refer to the set of all natural numbers.^ I’d be surprised to see natural log taken on a straight number (total assets or sales) — usually it’d be used for a growth factor or a series.
• Demystifying the Natural Logarithm (ln) | BetterExplained 1 February 2010 3:28 UTC betterexplained.com [Source type: General]

^ As you say, while it is “possible” to have a negative base, it’s likely not very useful because it only works for a small set of numbers.
• Demystifying the Natural Logarithm (ln) | BetterExplained 1 February 2010 3:28 UTC betterexplained.com [Source type: General]

^ Here the attribute “X-number” only refers to the fact that distinct sets of objects have distinct zeros.
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

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 aleph-null $(\aleph_0)$.^ As you say, while it is “possible” to have a negative base, it’s likely not very useful because it only works for a small set of numbers.
• Demystifying the Natural Logarithm (ln) | BetterExplained 1 February 2010 3:28 UTC betterexplained.com [Source type: General]

^ Cardinal numbers are the ones that are used to count the elements of a set.
• Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]

^ But induction is essential: the natural numbers are an infinite set - so if we want to be able to say anything about the entire set, then we need to be able to use that kind of reasoning to extend from the finite to the infinite.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

To be unambiguous about whether zero is included or not, sometimes an index "0" is added in the former case, and a superscript "*" or subscript "1" is added in the latter case:
$\mathbb{N}_0 = \{ 0, 1, 2, \ldots \}; \quad \mathbb{N}^* = \mathbb{N}_1 = \{ 1, 2, \ldots \}.$
(Sometimes, an index or superscript "+" is added to signify "positive". 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.)
.Some authors who exclude zero from the naturals use the terms natural numbers with zero, whole numbers, or counting numbers, denoted W, for the set of nonnegative integers.^ I, too, had been under the impression that zero was not a natural, or counting, number but, rather, the first in the set of whole numbers.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ Cardinal numbers are the ones that are used to count the elements of a set.
• Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]

^ Take the set of natural numbers N .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

.Others use the notation P for the positive integers if there is no danger of confusing this with the prime numbers.^ There are infinitely many prime numbers .
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ I was taught that natural numbers are the positive integers (i.e.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ Legendre's conjecture: There is a prime number between n2 and (n + 1)2 for every positive integer n.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

.Set theorists often denote the set of all natural numbers including zero by a lower-case Greek letter omega: ω.^ Take the set of natural numbers N .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ According to this convention, zero is not a natural number.
• Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]

^ We usually write N for the set of natural numbers.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

This stems from the identification of an ordinal number with the set of ordinals that are smaller.

## Algebraic properties

The addition and multiplication operations on natural numbers 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: for all natural numbers a and b, a + b = b + a and a × b = b × a.
• Existence of identity elements: for every natural number a, a + 0 = a and a × 1 = a.
• Distributivity for all natural numbers a, b, and c, a × (b + c)  =  (a × b) + (a × c)
• No zero divisors: if a and b are natural numbers such that a × b = 0   then a = 0 or b = 0

## 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.^ All of these theories subscribe to one or more basic tenets of natural law legal theory and are important to its development and influence.
• Natural Law [Internet Encyclopedia of Philosophy] 1 February 2010 3:28 UTC www.utm.edu [Source type: Original source]

^ I, too, had been under the impression that zero was not a natural, or counting, number but, rather, the first in the set of whole numbers.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ Addition is a function "+" from a pair of natural numbers to another natural number called their sum .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

.Here S should be read as "successor". This turns the natural numbers (N, +) into a commutative monoid with identity element 0, the so-called free monoid with one generator.^ Predecessor rule: 0 is not the successor of any natural number.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ Successor rule : For every natural number n there is exactly one other natural number called its successor s(n) .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ Location: Natural History -- Call Number: 581.529 R313 .

This monoid satisfies the cancellation property and can be embedded in a group. .The smallest group containing the natural numbers is the integers.^ The smallest group containing the natural numbers is the integers .
• Natural number - Psychology Wiki 1 February 2010 3:28 UTC psychology.wikia.com [Source type: Reference]
• WikiSlice 1 February 2010 3:28 UTC dev.laptop.org [Source type: Reference]

^ I was taught that natural numbers are the positive integers (i.e.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ What are the natural numbers or positive integers?
• Goalfinder - Integers, Natural and Whole Numbers ( Math Software ) - Animated Easy Science, Technology Software, Online Education, medical, K12 animation, & e-Learning 1 February 2010 3:28 UTC www.goalfinder.com [Source type: Academic]

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.^ Lemma :  Matrices constitute an Abelian group under addition.  We define subtraction in an analogous manner to that employed for the integers.
• Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]

^ Definition :  The reciprocal operation is defined as an indication of the multiplicative-inverse of a given number (a/b); that is 1/(a/b) = b/a.
• Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]

^ For example, addition and multiplication can be defined as compositions at two levels: (σ+τ)(f) = σ(f)∘τ(t), (σ×τ)(f) = (σ∘τ)(t).
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

.This turns (N*, ×) into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers.^ Consider how we establish whether two collections of physical object (two sets in the everyday sense) have the same number of elements.
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

^ The notion of prime number has been generalized in many different branches of mathematics.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ Prime ideals are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

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.^ Addition is a function "+" from a pair of natural numbers to another natural number called their sum .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ Many numbers occur in nature, and inevitably some of these are prime.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ In general, the properties of ℕ X depend on X, and only by setting restrictions on X can we interpret X-numbers as natural numbers.
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

.Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative.^ If the natural number a is divisible by the natural number b, then a is a multiple of b.

^ These properties of addition and multiplication make the natural numbers an instance of a commutative semiring .
• Natural number - Psychology Wiki 1 February 2010 3:28 UTC psychology.wikia.com [Source type: Reference]

^ Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative.
• Natural number - Psychology Wiki 1 February 2010 3:28 UTC psychology.wikia.com [Source type: Reference]
• WikiSlice 1 February 2010 3:28 UTC dev.laptop.org [Source type: Reference]

.The lack of additive inverses, which is equivalent to the fact that N is not closed under subtraction, means that N is not a ring.^ Definition .  A subset S of G  is said to be a subgroup of G if S is a group under the same operation as that of G.  Theorem .  A subset S of a group G  is a subgroup; if for any elements a and b of S the sum of a and the additive inverse of b is an element of S, namely, a + (-b) is in S. Theorem : Transitivity of the subgroup property .
• Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]

^ If we don't define the symbol "-0," you can write it to mean "the result of applying the operation of negation to 0," but you can't write it to mean "the symbol which represents the additive inverse of 0" unless we specifically define such a symbol.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ The symbol "-" serves at least 3 distinct purposes: to denote subtraction of one number from another, to denote the function of "taking the negative," and as part of a set of symbols which we use for the additive inverses of the positive integers (where it has no independent meaning).
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

.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.^ John Foster says that Natural numbers “are the ones we use to count things that are there.” But that’s his definition…his opinion.
• Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]

^ At my school, it is taught (in my math class) that the set of natural numbers starts at 1.
• Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]

^ Natural numbers as exponents of operations - a formalization of Wittgenstein’s definition Natural numbers as exponents of operations A formalization of Wittgenstein ’s definition .
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

For the remainder of the article, we write ab to indicate the product a × b, and we also assume the standard order of operations.
.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.^ We usually write N for the set of natural numbers.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ In this case there is only one X-number, and consequently PA2 is true.
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

^ The Bernoulli Numbers seem almost random in nature even though there are precise regular patterns that can be constructed to obtain their exact values.
• Sums of Powers of the Natural Numbers 1 February 2010 3:28 UTC members.bex.net [Source type: Reference]

.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 acbc.^ Author: Iverson, Ben & Elkins, Dr. Carl Natural arithmetic, number theory and geometry are the basis of all natural phenomena.
• SVP Catalog - Number and Number Theory 1 February 2010 3:28 UTC www.svpvril.com [Source type: FILTERED WITH BAYES]

^ Natural arithmetic, number theory and geometry is the basis of all natural phenomena.
• SVP Catalog - Number and Number Theory 1 February 2010 3:28 UTC www.svpvril.com [Source type: FILTERED WITH BAYES]

^ Author: Iverson, Ben Natural arithmetic, number theory and geometry is the basis of all natural phenomena.
• SVP Catalog - Number and Number Theory 1 February 2010 3:28 UTC www.svpvril.com [Source type: FILTERED WITH BAYES]

.An important property of the natural numbers is that they are well-ordered: every non-empty set of natural numbers has a least element.^ Take the set of natural numbers N .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ Hence, the natural numbers are totally ordered.
• Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]

^ Consider ordered pairs of natural numbers (a, b).
• Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]

The rank among well-ordered sets is expressed by an ordinal number; for the natural numbers this is expressed as "ω".
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.
.The number q is called the quotient and r is called the remainder of division of a by b.^ The resulting number is not divisible by any of the primes in the finite set we considered, because dividing by any of these would give a remainder of one.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

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.^ Main article: public key cryptography Several public-key cryptography algorithms, such as RSA, are based on large prime numbers (for example with 512 bits).
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ All prime numbers above 3 are of form 6n − 1 or 6n + 1, because all other numbers are divisible by 2 or 3.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

## 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.^ But there is a difference between the use of a hammer and the use of numbers.
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

^ Cardinal numbers are the ones that are used to count the elements of a set.
• Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]

^ Take the set of natural numbers N .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

.The set of natural numbers itself and any other countably infinite set has cardinality aleph-null ($\aleph_0$).
• Ordinal numbers "first", "second", "third" can be assigned to the elements of a totally ordered finite set, and also to the elements of well-ordered countably infinite sets like the set of natural numbers itself.^ The ordinal forms are First, second, third, fourth, .
• Ratio of natural numbers. Evolution of the real numbers. 1 February 2010 3:28 UTC www.themathpage.com [Source type: Original source]

^ Take the set of natural numbers N .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ The natural numbers have two forms, cardinal and ordinal .
• Ratio of natural numbers. Evolution of the real numbers. 1 February 2010 3:28 UTC www.themathpage.com [Source type: Original source]

.This can be generalized to ordinal numbers which describe the position of an element in a well-order set in general.^ Consider how we establish whether two collections of physical object (two sets in the everyday sense) have the same number of elements.
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

^ Whole numbers are the positive naturals; I've generally seen N 0 as a notation for the natural numbers without zero - that is, the whole numbers.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ In general, the properties of ℕ X depend on X, and only by setting restrictions on X can we interpret X-numbers as natural numbers.
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

.An ordinal number is also used to describe the "size" of a well-ordered set, in a sense different from cardinality: if there is an order isomorphism between two well-ordered sets they have the same ordinal number.^ But there is a difference between the use of a hammer and the use of numbers.
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

^ Cardinal numbers are the ones that are used to count the elements of a set.
• Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]

^ We use that same ordinal number to name the part.
• Ratio of natural numbers. Evolution of the real numbers. 1 February 2010 3:28 UTC www.themathpage.com [Source type: Original source]

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 well-ordered sets with cardinal number $\aleph_0$ have an ordinal number greater than ω.^ We are explaining how the ordinal numbers -- third, fourth, fifth, and so on -- name the parts of the cardinal numbers.
• Ratio of natural numbers. Evolution of the real numbers. 1 February 2010 3:28 UTC www.themathpage.com [Source type: Original source]

^ Cardinal numbers are the ones that are used to count the elements of a set.
• Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]

^ As regards to the relationship between the two definitions, the cardinal number of a finite set A could be defined within our formalism as follows.
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

For example,
$\omega^{\omega^{\omega6+42}\cdot1729+\omega^9+88}\cdot3+\omega^{\omega^\omega}\cdot5+65537$
has cardinality $\aleph_0$. The least ordinal of cardinality $\aleph_0$ (i.e., the initial ordinal) is ω.
.For finite well-ordered sets, there is 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.^ Cardinal numbers are the ones that are used to count the elements of a set.
• Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]

^ Take the set of natural numbers N .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ Hence, the natural numbers are totally ordered.
• Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]

.This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence.^ The digit use is distinguished by prefacing a digit sequence with an additional braille character, called a number sign, that indicates the change of semantics.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ For any natural number N larger than 1, the sequence (for the notation N! read factorial) .
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ Cardinal numbers are the ones that are used to count the elements of a set.
• Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]

.Other generalizations are discussed in the article on numbers.^ The other day, I was discussing a number of suggestions to improve Office’s spell-checker .
• Office Natural Language Team Blog 1 February 2010 3:28 UTC blogs.msdn.com [Source type: General]

^ Another formula is based on Wilson's theorem mentioned above, and generates the number two many times and all other primes exactly once.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ The American Mathematical Society apparently feels that the ontology of numbers and other mathematical objects is nontrivial enough to include many articles on the topic in their Mathematical Reviews .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

## Formal definitions

.Historically, the precise mathematical definition of the natural numbers developed with some difficulty.^ Natural gas and some worrying numbers .
• Natural gas and some worrying numbers | Energy Bulletin 1 February 2010 3:28 UTC www.energybulletin.net [Source type: FILTERED WITH BAYES]

^ In the nineteenth century, a set-theoretical definition of natural numbers was developed.
• WikiSlice 1 February 2010 3:28 UTC dev.laptop.org [Source type: Reference]

^ A natural number is a mathematical object .

.The Peano axioms state conditions that any successful definition must satisfy.^ Actuallly the Peano Axioms don't really provide a definition of the naturals.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ Definition :  A commutative ring with identity satisfies the fifth (commutative) axiom as well.
• Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]

^ Definition :  A division-ring (sometimes called a non-commutative field) satisfies all of the group axioms under multiplication.
• Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]

.Certain constructions show that, given set theory, models of the Peano postulates must exist.^ In fact no set of first order axioms can define any particular infinite model, because any consistent (first order) theory that has a model of (infinite) cardinality \kappa has a model of every cardinality greater than \kappa.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

### Peano axioms

.The Peano axioms give a formal theory of the natural numbers.^ What Marc is using is a variant of the Peano Axioms to define the natural numbers.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ The natural numbers come from the Peano axioms.
• Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]

^ The study of prime numbers is part of number theory, the branch of mathematics which encompasses the study of natural numbers.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

The axioms are:
.
• There is a natural number 0.
• Every natural number a has a natural number successor, denoted by S(a).^ Successor rule : For every natural number n there is exactly one other natural number called its successor s(n) .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ Additive Inverse: For any natural number n other than zero, n, there is exactly one number -n which not a natural number, and which called the additive inverse of n, where n + -n = 0 .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ John Foster says that Natural numbers “are the ones we use to count things that are there.” But that’s his definition…his opinion.
• Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]

.Intuitively, S(a) is a+1.
• There is no natural number whose successor is 0.
• Distinct natural numbers have distinct successors: if ab, then S(a) ≠ S(b).
• If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers.^ In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ Because all non-prime numbers can be decomposed into a product of underlying primes, then either this resultant number is prime itself, or there is a prime number or prime numbers which the resultant number could be decomposed into but are not in the original finite set of primes.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

^ However there is no evidence to suggest that starfish have 5 arms because 5 is a prime number.
• The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]

.(This postulate ensures that the proof technique of mathematical induction is valid.^ Proof:  Its contra positive is a subset of the natural numbers, which does not possess a least element, is null.  Hence, it obviously is equivalent to the Mathematical Induction axiom.  QED. .
• Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]

^ Proof by Mathematical Induction.
• Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]

)
.It should be noted that the "0" in the above definition need not correspond to what we normally consider to be the number zero.^ It should be noted that the "0" in the above definition need not correspond to what we normally consider to be the number zero.
• WikiSlice 1 February 2010 3:28 UTC dev.laptop.org [Source type: Reference]

^ This particular category is considered the"business minded numbers."  Your 2 Life Path is the number you should be concentrating on .

^ With this definition, it was more convenient to include zero (corresponding to the empty set) as a natural number.
• WikiSlice 1 February 2010 3:28 UTC dev.laptop.org [Source type: Reference]

."0" simply means some object that when combined with an appropriate successor function, satisfies the Peano axioms.^ Assuming the axiom of infinity , this definition can be shown to satisfy the Peano axioms.
• Natural number - Psychology Wiki 1 February 2010 3:28 UTC psychology.wikia.com [Source type: Reference]

^ As pointed out by Bertrand Russell in 'Introduction to Mathematical Philosophy' (and perhaps others), '0 and 'number' and 'successor' are primitive terms in Peano's axioms and so aren't really defined.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ Now if you really want to blow some people's minds, you should talk about the incompleteness of the Peano axioms and the existence of nonstandard models of arithmetic .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

.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.^ There is no doubt whatever that she is entitled to do so, and I think everybody will agree too that actinium K is no suitable name for the element, but only for that particular isotope.
• Francium (Atomic Number 87), the Last Discovered Natural Element 1 February 2010 3:28 UTC chemeducator.org [Source type: FILTERED WITH BAYES]

^ It is only natural that the symbol of the element conveys the same impression as the full name, and personally, I do not think that the life-time matters [28].
• Francium (Atomic Number 87), the Last Discovered Natural Element 1 February 2010 3:28 UTC chemeducator.org [Source type: FILTERED WITH BAYES]

^ I think it is a fairly general rule in chemistry that as a symbol the first letter, or the two first letters, of the name are taken, unless they are already given to another element.
• Francium (Atomic Number 87), the Last Discovered Natural Element 1 February 2010 3:28 UTC chemeducator.org [Source type: FILTERED WITH BAYES]

.For example, the natural numbers starting with one also satisfy the axioms, if the symbol 0 is interpreted as the natural number 1, the symbol S(0) as the number 2, etc.^ The natural numbers come from the Peano axioms.
• Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]

^ One is a natural number: 1 belongs to N. If a is a natural number; then so is its successor: If a belongs to N; then s(a) belongs to N. The Mathematical Induction axiom.
• Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]

^ Additive Inverse: For any natural number n other than zero, n, there is exactly one number -n which not a natural number, and which called the additive inverse of n, where n + -n = 0 .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

In fact, in Peano's original formulation, the first natural number was 1.

### 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 $S(a) = a \cup \{a\}$ for every set a. 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.
If the set of all natural numbers exists, then it satisfies the Peano axioms.
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}}} = {{ }, {{ }}, {{ }, {{ }}}}
• $n = \{0, 1, 2, \ldots, n-2, n-1\} = \{0, 1, 2, \ldots, n-2\} \cup \{n-1\} = (n-1) \cup \{n-1\}$
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 nm (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.^ Although the standard construction is useful, it is not the only possible construction.
• Natural number - Psychology Wiki 1 February 2010 3:28 UTC psychology.wikia.com [Source type: Reference]
• WikiSlice 1 February 2010 3:28 UTC dev.laptop.org [Source type: Reference]

^ Note that in the above-quoted treatment, natural numbers were used not only for illustration as data items but also in the construct ‘a n ’ as exponents.
• Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]

^ As you say, while it is “possible” to have a negative base, it’s likely not very useful because it only works for a small set of numbers.
• Demystifying the Natural Logarithm (ln) | BetterExplained 1 February 2010 3:28 UTC betterexplained.com [Source type: General]

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 ∪ {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.^ Take the set of natural numbers N .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ We usually write N for the set of natural numbers.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ I, too, had been under the impression that zero was not a natural, or counting, number but, rather, the first in the set of whole numbers.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

[5][6] .This may appear circular, but can be made rigorous with care.^ This may appear circular, but can be made rigorous with care.
• Natural number - Psychology Wiki 1 February 2010 3:28 UTC psychology.wikia.com [Source type: Reference]

Define 0 as {{ }} (clearly the set of all sets with 0 elements) and define S(A) (for any set A) as {x ∪ {y} | xAyx } (see set-builder notation). Then 0 will be the set of all sets with 0 elements, 1 = S(0) will be the set of all sets with 1 element, 2 = S(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 S (that is, if the set contains an element n, it also contains S(n)).^ Take the set of natural numbers N .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ We usually write N for the set of natural numbers.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ I, too, had been under the impression that zero was not a natural, or counting, number but, rather, the first in the set of whole numbers.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

.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 relatively consistent) and in some systems of type theory.^ As you say, while it is “possible” to have a negative base, it’s likely not very useful because it only works for a small set of numbers.
• Demystifying the Natural Logarithm (ln) | BetterExplained 1 February 2010 3:28 UTC betterexplained.com [Source type: General]

^ People from more number-theory type backgrounds were taught the one-based definition.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ In fact no set of first order axioms can define any particular infinite model, because any consistent (first order) theory that has a model of (infinite) cardinality \kappa has a model of every cardinality greater than \kappa.
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

## Notes

1. ^ Calculus I by Jerrold E. Marsden and Alan Weinstein, page 15
2. ^ "... a tablet found at Kish ... thought to date from around .700 BC, uses three hooks to denote an empty place in the positional notation.^ BC , uses three hooks to denote an empty place in the positional notation.
• Natural number - Psychology Wiki 1 February 2010 3:28 UTC psychology.wikia.com [Source type: Reference]

^ The notation N_0 (where the "_" denotes subscript) is totally unhelpful because it's used to mean "the opposite of whatever N means" both by people who include 0 in N and by people who don't include 0 in N. .
• Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]

^ A zero digit had been used in place-value notation as early as 700 BC by the Babylonians, but it was never used as a final element.
• Natural number - Psychology Wiki 1 February 2010 3:28 UTC psychology.wikia.com [Source type: Reference]

Other tablets dated from around the same time use a single hook for an empty place. [1]"
3. ^ This convention is used, for example, in Euclid's Elements, see Book VII, definitions 1 and 2.
4. ^ This is common in texts about Real analysis. See, for example, Carothers (2000) p.3 or Thomson, Bruckner and Bruckner (2000), p.2.
5. ^ Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl (1884). Breslau.
6. ^ Whitehead, Alfred North, and Bertrand Russell. Principia Mathematica, 3 vols, Cambridge University Press, 1910, 1912, and 1913. Second edition, 1925 (Vol. 1), 1927 (Vols 2, 3). Abridged as Principia Mathematica to *56, Cambridge University Press, 1962.

## References

• Edmund Landau, Foundations of Analysis, Chelsea Pub Co. ISBN 0-8218-2693-X.
• Richard Dedekind, Essays on the theory of numbers, Dover, 1963, ISBN 0486210103 / Kessinger Publishing, LLC , 2007, ISBN 054808985X
• N. L. Carothers. Real analysis. .Cambridge University Press, 2000. ISBN 0521497566
• Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner.^ In Out of the Shadows: Contributions of 20 th Century Women to Physics ; Byers, N.; Williams, G., Eds.; Cambridge University Press: Cambridge, England, 2005; pp 371384.
• Francium (Atomic Number 87), the Last Discovered Natural Element 1 February 2010 3:28 UTC chemeducator.org [Source type: FILTERED WITH BAYES]

Elementary real analysis. ClassicalRealAnalysis.com, 2000. ISBN 0130190756

# Simple English

Natural numbers , also called Counting numbers, are the numbers used for counting things. Natural numbers are positive integers (whole numbers that are more than 0). They are 1, 2, 3, 4, 5... and so on until infinity. Infinity is not a natural number. They could also be said to be the set of all possible numbers of elements in any finite set.

Natural numbers got its name because it is found naturally in nature. Therefore, 0, -1, -2, -3, -4... are not natural numbers.

## Notation

$\mathbf\left\{N\right\}$ or $\mathbb\left\{N\right\}$ is the way to write the set of all natural numbers. Because some people say 0 is a natural number, and some people say it is not, people use the following symbols to talk about the natural numbers:

Symbol Meaning
$\mathbb\left\{N\right\}^+$ Positive numbers, without zero
$\mathbb\left\{N\right\}^*$ Positive numbers without zero
$\mathbb\left\{N\right\}_0$ Positive numbers, with zero
$\mathbb\left\{N\right\}_\left\{>0\right\}$ Positive numbers without zero
$\mathbb\left\{N\right\} \setminus \\left\{0\\right\}$ Positive numbers without zero

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# Citable sentences

Up to date as of December 19, 2010

Here are sentences from other pages on Natural number, which are similar to those in the above article.