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Encyclopedia > Natural number

In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i.e the positive integers or the counting numbers) or an element of the set {0, 1, 2, 3, ...} (i.e. the non-negative integers). The former is generally used in number theory, while the latter is preferred in mathematical logic, set theory and computer science. See below for a formal definition. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Look up one in Wiktionary, the free dictionary. ... This article does not cite any references or sources. ... Look up three in Wiktionary, the free dictionary. ... A negative number is a number that is less than zero, such as &#8722;3. ... The integers are commonly denoted by the above symbol. ... 0 (zero) is both a number and a numerical digit used to represent that number in numerals. ... A negative number is a number that is less than zero, such as &#8722;3. ... 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 of problems that arise from their study. ... Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ...

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 with one for the first object and proceeding with an injective function from the remaining objects to the natural numbers... 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. ...

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. 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 a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... 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 of problems that arise from their study. ... 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 pure mathematics concerning the study of discrete (and usually finite) objects. ...

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

Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ...

## History of natural numbers and the status of zero GA_googleFillSlot("encyclopedia_square");

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, or a word in a natural language that represents a number. ... Babylonia was a state in the south part of Mesopotamia (in modern Iraq), combining the territories of Sumer and Akkad. ... 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 common ratio, called the base or radix of that numeral system. ... A section of the Papyrus of Ani showing cursive hieroglyphs. ... Map of Karnak, showing major temple complexes Interior of Temple First pylon of precinct of Amun viewed from the west Al-Karnak (Arabic Ø§Ù„ÙƒØ±Ù†Ùƒ, in Ancient Egypt was named Ipet Sut, the most venerated place) is a small village in Egypt, located on the banks of the River Nile some 2. ... (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... This article does not cite any references or sources. ...

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.[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. 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," nullus, was employed. 0 (zero) is both a number and a numerical digit used to represent that number in numerals. ... 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. ... Monument 1, one of the four Olmec colossal heads at La Venta. ... The Maya civilization is a Mesoamerican civilization, noted for the only known fully developed written language of the pre-Columbian Americas, as well as its spectacular art, monumental architecture, and sophisticated mathematical and astronomical systems. ... (2nd millennium BC - 1st millennium BC - 1st millennium) The 1st century BC started on January 1, 100 BC and ended on December 31, 1 BC. An alternative name for this century is the last century BC. The AD/BC notation does not use a year zero. ... Location of Mesoamerica in the Americas. ... Brahmagupta (à¤¬à¥à¤°à¤¹à¥à¤®à¤—à¥à¤ªà¥à¤¤) (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. ... Topics in Christianity Movements Â· Denominations Ecumenism Â· Preaching Â· Prayer Music Â· Liturgy Â· Calendar Symbols Â· Art Â· Criticism Important figures Apostle Paul Â· Church Fathers Constantine Â· Athanasius Â· Augustine Anselm Â· Aquinas Â· Palamas Â· Wycliffe Tyndale Â· Luther Â· Calvin Â· Wesley Arius Â· Marcion of Sinope Pope Â· Archbishop of Canterbury Patriarch of Constantinople Christianity Portal This box:      Easter, the Sunday of... 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. abstraction in general. ... An entity is something that has a distinct, separate existence, though it need not be a material existence. ... Pythagoras of Samos (Greek: ; between 580 and 572 BCâ€“between 500 and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ... Archimedes of Syracuse (Greek: c. ... Location of Mesoamerica in the Americas. ...

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. ... The empty set is the set containing no elements. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Logic (from Classical Greek Î»ÏŒÎ³Î¿Ï‚ logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... 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 of problems that arise from their study. ...

## 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. 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 ($aleph_0$). An example of blackboard bold letters. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In set theory, an infinite set is a set that is not a finite set. ... In mathematics the term countable set is used to describe the size of a set, e. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. ...

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:

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,... }, at least in European literature. The notation "*", however, is standard for nonzero or rather invertible elements.) This article is about the term superscript as used in typography. ... 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. ...

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. When this notation is used, zero is explicitly included as a natural number. Look up Î©, Ï‰ in Wiktionary, the free dictionary. ...

## Formal definitions

Main article: Set-theoretic definition of natural numbers

Historically, the precise mathematical definition of the natural numbers developed with some difficulty. The Peano postulates state conditions that any successful definition must satisfy. Certain constructions show that, given set theory, models of the Peano postulates must exist. Several ways have been proposed to define the natural numbers using set theory. ... 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). ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ...

### Peano axioms

• 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 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. (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. 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 satify the axioms. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...

### 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: Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ...

We set 0 := { }, the empty set,
and define S(a) = a ∪ {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,…,n−2,n−1} = {0,1,2,…,n−2} ∪ {n−1} = (n−1) ∪ {n−1}
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 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.

The empty set is the set containing 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 Peano axioms (or Peano postulates) are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of the natural numbers. ... It has been suggested that this article or section be merged into Logical biconditional. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...

#### 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.[citation needed] 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 ${x cup {y} mid x in A wedge y notin x}$. 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. Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 - July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ... Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, (18 May 1872 â€“ 2 February 1970), was a British philosopher, logician, mathematician, advocate for social reform, and pacifist. ... This article or section is in need of attention from an expert on the subject. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, or axiom schema of separation, or axiom schema of restricted comprehension, is a schema of axioms in Zermelo-Fraenkel set theory. ... In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. ... At the broadest level, type theory is the branch of mathematics and logic that first creates a hierarchy of types, then assigns each mathematical (and possibly other) entity to a type. ...

## 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 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. ... In mathematics, the idea of a free object is one of the basic concepts 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. ... This picture illustrates how the hours on a clock form a group under modular addition. ... The integers are commonly denoted by the above symbol. ...

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. 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, multiplication is an elementary arithmetic operation. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 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 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 arithmetic and algebra, when a number or expression is both preceded and followed by a binary operation, a rule is required for which operation should be applied first. ...

Furthermore, one defines a total order on the natural numbers by writing ab 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 ab, then a + cb + c and acbc. An important property of the natural numbers is that they are well-ordered: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an ordinal number; for the natural numbers this is expressed as "ω". In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... The feasible regions of linear programming are defined by a set of inequalities. ... 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. ... In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ...

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 In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ...

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. In mathematics, a quotient is the end result of a division problem. ... In mathematics, the result of the division of two integers usually cannot be expressed with an integer quotient, unless a remainder â€”an amount left overâ€” is also acknowledged. ... 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. ... In number theory, the Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (GCD) of two elements of any Euclidean domain (for example, the integers). ...

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 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. ...

## Generalizations

Two generalizations of natural numbers arise from the two uses:

• A natural number can be used to express the size of a finite set; more generally a cardinal number is a measure for the size of a set also suitable for infinite sets; this refers to a concept of "size" such that if there is a bijection between two sets they have the same size. The set of natural numbers itself and any other countably infinite set has cardinality 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. This can be generalized to ordinal numbers which describe the position of an element in a well-order set in general. 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. 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.

$aleph_0$ and ωhave 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). Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... Two sets A and B are said to be equinumerous if they have the same cardinality, i. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality â€“ one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. ... In linguistics, ordinal numbers are the words representing the rank of a number with respect to some order, in particular chronological order or position: first, second, third, etc. ... In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ... In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets. ... The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. ...

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. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence. In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...

Other generalizations are discussed in the article on numbers. For other uses, see Number (disambiguation). ...

## 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]"
• Edmund Landau, Foundations of Analysis, Chelsea Pub Co. ISBN 0-8218-2693-X.

Edmund Georg Hermann (Yehezkel) Landau (February 14, 1877 â€“ February 19, 1938) was a German Jew mathematician and author of over 250 papers on number theory. ...

Results from FactBites:

 Natural number - Wikipedia, the free encyclopedia (1844 words) 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"). Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. Two generalizations of natural numbers arise from the two uses: ordinal numbers are used to describe the position of an element in an ordered sequence and cardinal numbers are used to specify the size of a given set.
 Number - Wikipedia, the free encyclopedia (3699 words) The arithmetical operations of numbers, such as addition, subtraction, multiplication and division, are generalized in the branch of mathematics called abstract algebra, the study of abstract number systems such as groups, rings and fields. This number is denoted by i, a symbol assigned by Leonhard Euler. The existence of complex numbers was not completely accepted until the geometrical interpretation had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion.
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