Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), or they can be used for ordering ("this is the 3rd largest city in the country"). Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. Problems concerning counting, such as Ramsey theory, are studied in combinatorics. A negative number is a number that is less than zero, such as −3. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... Look up one in Wiktionary, the free dictionary 1 (one) is a number, numeral, and glyph. ... For the Austin Powers character, see Number 2(Austin Powers 2 (two) is a number, numeral, and glyph. ... 3 (three) is a number, numeral, and glyph. ... 4 (four) is a number, numeral, and glyph. ... A negative number is a number that is less than zero, such as −3. ... 0 (zero) or nought is both a number and a numeral. ... Look up one in Wiktionary, the free dictionary 1 (one) is a number, numeral, and glyph. ... For the Austin Powers character, see Number 2(Austin Powers 2 (two) is a number, numeral, and glyph. ... 3 (three) is a number, numeral, and glyph. ... 4 (four) is a number, numeral, and glyph. ... Counting is the mathematical action of adding (or subtracting) one, usually to find out how many objects there are or to set aside a desired number of objects (starting with one for the first object and proceeding with a one-to-one correspondence); however, counting is also used (primarily by... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... Ramsey theory, named for Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear. ... Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria. ...

History of natural numbers and the status of zero

The natural numbers presumably 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. 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. The ancient Egyptians had a system of numerals with distinct hieroglyphs for 1, 10, and all the powers of 10 up to one million. 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 numeral is a symbol or group of symbols that represents a number. ... Babylonia was an ancient state in Mesopotamia (in modern Iraq), combining the territories of Sumer and Akkad. ... A numeral is a symbol or group of symbols that represents a number. ... A hieroglyph is one part of an ideographic writing system that is often found carved in stone. ... Entrance to Precinct of Amon-Re of the Karnak Temple Complex This article is about the village and pharaonic temple complex in Egypt. ... (Redirected from 1500 BC) Centuries: 17th century BC - 16th century BC - 15th century BC Decades: 1550s BC 1540s BC 1530s BC 1520s BC 1510s BC - 1500s BC - 1490s BC 1480s BC 1470s BC 1460s BC 1450s BC Events and Trends Stonehenge built in Wiltshire, England The element Mercury has been... I.M. Peis Louvre Pyramid: the entrance to the galleries lies below the glass pyramid The Louvre Museum (Musée du Louvre) in Paris, France, is one of the largest and most famous museums in the world. ...

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 it was never used as a final element.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 concept as used in modern times originated with the Indian mathematician Brahmagupta in 628 AD. 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. Instead, the Latin word for "nothing," nullae, was employed. 0 (zero) or nought is both a number and a numeral. ... In mathematics and computer science, a numerical digit is a symbol, e. ... Centuries: 9th century BC - 8th century BC - 7th century BC Decades: 750s BC 740s BC 730s BC 720s BC 710s BC - 700s BC - 690s BC 680s BC 670s BC 660s BC 650s BC Events and Trends 708 BC - Spartan immigrants found Taras (Tarentum, the modern Taranto) colony in southern Italy. ... The Olmec were an ancient people living in the tropical lowlands of south-central Mexico, roughly in what are the modern-day states of Veracruz and Tabasco on the Isthmus of Tehuantepec. ... The Maya are people of southern Mexico and northern Central America (Guatemala, Belize, western Honduras, and El Salvador) with some 3,000 years of history. ... (2nd century BC - 1st century BC - 1st century - other centuries) The 1st century BC starts on January 1, 100 BC and ends on December 31, 1 BC. An alternative name for this century is the last century BC. (2nd millennium BC - 1st millennium BC - 1st millennium) // Events The Roman Republic... Mesoamerica is the region extending from central Mexico south to the northwestern border of Costa Rica that gave rise to a group of stratified, culturally related agrarian civilizations spanning an approximately 3,000-year period before the African discovery of the New World by Columbus. ... Brahmagupta (ब्रह्मगुप्त) (598_668) was an Indian mathematician and astronomer. ... Events Khusro II of Persia overthrown Pippin of Landen becomes Mayor of the Palace Brahmagupta writes the Brahmasphutasiddhanta Births Deaths Empress Suiko of Japan Theodelinda, queen of the Lombards Categories: 628 ... Computus (Latin for computation) is the calculation of the date of Easter in the Christian calendar. ... Easter is the most important religious holiday of the Christian liturgical year, observed in March, April, or May to celebrate the resurrection of Jesus from the dead after his death by crucifixion (see Good Friday), which Christians believe happened at about this time of year around AD 30-33. ... Dionysius Exiguus (Dennis the Little, meaning humble) (c. ... Events Bernicia settled by the Angles Ethiopia conquers Yemen The Daisan river, a tributary of the Euphrates, floods Edessa and within a couple of hours fills the entire city except for the highest parts. ... The system of Roman numerals is a numeral system originating in ancient Rome, and was adapted from Etruscan numerals. ...

The first systematic study of numbers as abstractions (that is, as abstract entities) is usually credited to the Greek philosophers Pythagoras and Archimedes. However, independent studies also occurred at around the same time in India, China, and Mesoamerica. // An abstraction is an idea, concept, or word which defines the phenomena which make up the concrete events or things which the abstraction refers to, the referents. ... An entity is something that has a distinction, separate existence, though it need not be a material existence. ... Pythagoras (582 BC – 496 BC, Greek: Πυθαγόρας) was an Ionian mathematician and philosopher, known best for formulating the Pythagorean theorem. ... Archimedes (Greek: ΑΡΧΙΜΗΔΗΣ) (287 BC–212 BC) was a Greek mathematician, astronomer, philosopher, physicist and engineer born in the Greek seaport colony of Syracuse. ... Mesoamerica is the region extending from central Mexico south to the northwestern border of Costa Rica that gave rise to a group of stratified, culturally related agrarian civilizations spanning an approximately 3,000-year period before the African discovery of the New World by Columbus. ...

In the nineteenth century, a set-theoretical definition of natural numbers was developed. With this definition, it was more convenient to include zero (corresponding to the empty set) as a natural number. This convention is followed by set theorists, logicians, and computer scientists. Other mathematicians, primarily number theorists, often prefer to follow the older tradition and exclude zero from the natural numbers. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Look up Definition in Wiktionary, the free dictionary For alternative meanings see definition (disambiguation) A definition may be a statement of the essential properties of a certain thing, or a statement of equivalence between a term and that terms meaning. ... In mathematics, the empty set is the set with no elements. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... It has been suggested that this article or section be merged with College logic. ... Wikibooks Wikiversity has more about this subject: School of Computer Science Open Directory Project: Computer Science Collection of Computer Science Bibliographies Belief that title science in computer science is inappropriate Categories: Computer science ... Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ...

The term whole number is used informally by some authors for an element of the set of integers, the set of non-negative integers, or the set of positive integers.

Notation

Mathematicians use N or (an N in blackboard bold) to refer to the set of all natural numbers. This set is infinite but countable by definition. Blackboard bold is a style of typeface often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical, or near-vertical lines) are doubled. ... In mathematics, a set can be thought of as any well-defined collection of things considered as a whole. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... In mathematics the term countable set is used to describe the size of a set, e. ...

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:

N = N₀ = { 0, 1, 2, ... } ; N^* = { 1, 2, ... }.

(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,... } = N, at least in European literature. The notation "*", however, is quite standard for nonzero or rather invertible elements.) In mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. ...

Less frequently, W or is used for the set of "whole numbers", which are sometimes identified with the natural numbers as defined here, sometimes with the integers (in which case N = W₊). The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

Formal definitions

Peano axioms

The precise mathematical definition of the natural numbers has not been easy. The Peano postulates state conditions that any successful definition must satisfy: In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ...

• There is a natural number 0.
• Every natural number a has a natural number successor, denoted by S(a).
• There is no natural number whose successor is 0.
• Distinct natural numbers have distinct successors: if a ≠ b, 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. (This postulate ensures that the proof technique of mathematical induction is valid.)

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. There are many systems that satisfy these axioms, including the natural numbers (either starting from zero or one). Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence. ...

The standard construction

A standard construction in set theory is to define the natural numbers as follows: Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

We set 0 := { }

and define S(a) = a U {a} for all a.

The set of natural numbers is then defined to be the intersection of all sets containing 0 which are closed under the successor function.

Assuming the axiom of infinity, this definition can be shown to satisfy 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}}} = {{ }, {{ }}, {{ }, {{ }}}}

and so on. When you see a natural number 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) iff n is a subset of m.

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

In mathematics, the empty set is the set with no elements. ... 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 of Zermelo-Fraenkel set theory. ... In mathematics, the empty set is the set with no elements. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...

Other constructions

Although this particular 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.

For the rest of this article, we follow the standard construction described first 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. This turns the natural numbers (N, +) into a commutative monoid with identity element 0, the so-called free monoid with one generator. This monoid satisfies the cancellation property and can therefore be embedded in a group. The smallest group containing the natural numbers is the integers. Addition of natural numbers is the most basic arithmetic operation. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... 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 set. ... The idea of a free object in mathematics is one of the basics of abstract algebra. ... In mathematics, an element a in a magma (M,*) has the left cancellation property (or is left-cancellative) if for all b and c in M, a * b = a * c always implies b = c. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

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

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 commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers. 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. In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In abstract algebra, a semiring is an algebraic structure, similar to a ring, but without additive inverses. ...

If we interpret the natural numbers as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that 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. Oh dear, looks like its time for a revert war!! ...

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. 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 total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In mathematics, a well-order (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. ...

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 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. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... The Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (gcd) of two integers. ...

Generalizations

Two generalizations of natural numbers arise from the two uses: ordinal numbers are used to describe the position of an element in a ordered sequence and cardinal numbers are used to specify the size of a given set. Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... This is a page about mathematics. ... In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... In mathematics, a set can be thought of as any well-defined collection of things considered as a whole. ...

For finite sequences or finite sets, both of these properties are embodied in the natural numbers. In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...

Other generalizations are discussed in the article on numbers. A number is an abstract entity used originally to describe quantity. ...

Footnote

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] Kish (Tall al-Uhaymir) was an ancient city of Sumer, now in central Iraq. ... Centuries: 9th century BC - 8th century BC - 7th century BC Decades: 750s BC 740s BC 730s BC 720s BC 710s BC - 700s BC - 690s BC 680s BC 670s BC 660s BC 650s BC Events and Trends 708 BC - Spartan immigrants found Taras (Tarentum, the modern Taranto) colony in southern Italy. ...