Number
21690
224992069
2008-07-11T10:27:36Z
Gwernol
266416
Reverted edits by [[Special:Contributions/122.106.87.238|122.106.87.238]] ([[User talk:122.106.87.238|talk]]) to last version by CrazyChemGuy
{{otheruses}}
A '''number''' is an [[abstract object]], [[Type-token distinction|token]]s of which are [[symbol]]s used in [[counting]] and [[measurement|measuring]]. A symbol which represents a number is called a [[Number names|numeral]], but in common usage the word number is used for both the abstract object and the symbol. In addition to their use in counting and measuring, numerals are often used for labels ([[telephone number]]s), for ordering ([[serial number]]s), and for codes ([[ISBN]]s). In [[mathematics]], the definition of number has been extended over the years to include such numbers as [[0 (number)|zero]], [[negative numbers]], [[rational numbers]], [[irrational numbers]], and [[complex numbers]]. As a result, there is no one encompassing definition of number and the concept of number is open for further development.
Certain procedures which input one or more numbers and output a number are called numerical [[operation (mathematics)|operations]]. [[Unary operator|Unary operations]] input a single number and output a single number. For example, the successor operation adds one to an integer: the successor of 4 is 5. More common are [[binary operator|binary operations]] which input two numbers and output a single number. Examples of binary operations include [[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[exponentiation]]. The study of numerical operations is called [[arithmetic]].
The branch of [[mathematics]] that studies structures of number systems such as [[group (mathematics)|group]]s, [[ring (algebra)|ring]]s and [[field (mathematics)|field]]s is called [[abstract algebra]].
==Types of numbers==
Numbers can be classified into [[set]]s, called [[number system]]s. (For different methods of expressing numbers with symbols, such as the [[Roman numerals]], see [[numeral system]]s.)
=== Natural numbers===
The most familiar numbers are the '''[[natural number]]s''' or counting numbers: one, two, three, ... . Some people also include zero in the natural numbers; however, others do not.
In the [[base ten]] number system, in almost universal use today for arithmetic operations, the symbols for natural numbers are written using ten [[Numerical digit|digit]]s: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this base ten system, the rightmost digit of a natural number has a place value of one, and every other digit has a place value ten times that of the place value of the digit to its right. The symbol for the set of all natural numbers is '''N''', also written [[Blackboard bold|<math>\mathbb{N}</math>.]]
In [[set theory]], which is capable of acting as an axiomatic foundation for modern mathematics, natural numbers can be represented by classes of equivalent sets. For instance, the number 3 can be represented as the class of all sets that have exactly three elements. Alternatively, in [[Peano Arithmetic]], the number 3 is represented as sss0, where s is the "successor" function. Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols 3 times.
===Integers===<!-- This section is linked from [[Kronecker delta]] -->
'''[[Negative number]]s''' are numbers that are less than zero. They are the opposite of positive numbers. For example, if a positive number indicates a bank deposit, then a negative number indicates a withdrawal of the same amount. Negative numbers are usually written by writing a negative sign (also called a minus sign) in front of the number they are the opposite of. Thus the opposite of 7 is written −7. When the [[set]] of negative numbers is combined with the natural numbers and zero, the result is the set of integer numbers, also called '''[[integer]]s''', '''Z''' (German ''Zahl'', plural ''Zahlen''), also written [[Blackboard bold|<math>\mathbb{Z}</math>.]]
===Rational numbers===
A '''[[rational number]]''' is a number that can be expressed as a [[Fraction (mathematics)|fraction]] with an integer [[numerator]] and a non-zero natural number [[denominator]]. The fraction ''m''/''n'' or
:<math>m \over n \, </math>
represents ''m'' equal parts, where ''n'' equal parts of that size make up one whole. Two different fractions may correspond to the same rational number; for example 1/2 and 2/4 are equal, that is:
: <math>{1 \over 2} = {2 \over 4}\,</math>.
If the [[absolute value]] of ''m'' is greater than ''n'', then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or zero. The set of all rational numbers includes the integers, since every integer can be written as a fraction with denominator 1. For example −7 can be written −7/1. The symbol for the rational numbers is '''Q''' (for ''[[quotient]]''), also written [[Blackboard bold|<math>\mathbb{Q}</math>.]]
===Real numbers===
The '''[[real numbers]]''' include all of the measuring numbers. Real numbers are usually written using [[decimal]] numerals, in which a decimal point is placed to the right of the digit with place value one. Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left. Thus
:<math>123.456\,</math>
represents 1 hundred, 2 tens, 3 ones, 4 tenths, 5 hundredths, and 6 thousandths. In saying the number, the decimal is read "point", thus: "one two three point four five six". In the US and UK and a number of other countries, the decimal point is represented by a [[Full stop|period]], whereas in continental Europe and certain other countries the decimal point is represented by a [[Comma (punctuation)|comma]]. Zero is often written as 0.0 when necessary to indicate that it is to be treated as a real number rather than as an integer. Negative real numbers are written with a preceding [[minus sign]]:
:<math>-123.456\,</math>.
Every rational number is also a real number. To write a fraction as a decimal, divide the numerator by the denominator. It is not the case, however, that every real number is rational. If a real number cannot be written as a fraction of two integers, it is called [[irrational number|irrational]]. A decimal that can be written as a fraction either ends (terminates) or forever repeats, because it is the answer to a problem in division. Thus the real number 0.5 can be written as 1/2 and the real number 0.333... (forever repeating threes) can be written as 1/3. On the other hand, the real number π ([[pi]]), the ratio of the [[circumference]] of any circle to its [[diameter]], is
:<math>\pi = 3.14159265358979...\,</math>.
Since the decimal neither ends nor forever repeats, it cannot be written as a fraction, and is an example of an irrational number. Other irrational numbers include
:<math>\sqrt{2} = 1.41421356237 ...\,</math>
(the [[square root of 2]], that is, the positive number whose square is 2).
Just as fractions can be written in more than one way, so too can decimals. For example, if we multiply both sides of the equation
:<math>1/3 = 0.333...\,</math>
by three, we discover that
:<math>1 = 0.999...\,</math>.
Thus 1.0 and [[0.999...]] are two different decimal numerals representing the natural number 1. There are infinitely many other ways of representing the number 1, for example 2/2, 3/3, 1.00, 1.000, and so on.
Every real number is either rational or irrational. Every real number corresponds to a point on the [[number line]]. The real numbers also have an important but highly technical property called the [[least upper bound]] property. The symbol for the real numbers is '''R''' or <math>\mathbb{R}</math>.
When a real number represents a [[measurement]], there is always a [[margin of error]]. This is often indicated by [[rounding]] or [[truncate|truncating]] a decimal, so that digits that suggest a greater accuracy than the measurement itself are removed. The remaining digits are called [[significant digits]]. For example, measurements with a ruler can seldom be made without a margin of error of at least 0.01 meters. If the sides of a [[rectangle]] are measured as 1.23 meters and 4.56 meters, then multiplication gives an area for the rectangle of 5.6088 square meters. Since only the first two digits after the decimal place are significant, this is usually rounded to 5.61.
In [[abstract algebra]], the real numbers are up to isomorphism uniquely characterized by being the only [[completeness (order theory)|complete]] [[ordered field]]. They are not, however, an [[algebraically closed field]].
===Complex numbers===
Moving to a greater level of abstraction, the real numbers can be extended to the '''[[complex number]]s'''. This set of numbers arose, historically, from the question of whether a negative number can have a [[square root]]. This led to the invention of a new number: the square root of negative one, denoted by ''[[imaginary unit|i]]'', a symbol assigned by [[Leonhard Euler]], and called the [[imaginary unit]]. The complex numbers consist of all numbers of the form
:<math>\,a + b i</math>
where ''a'' and ''b'' are real numbers. In the expression ''a'' + ''bi'', the real number ''a'' is called the ''real part'' and ''bi'' is called the ''imaginary part''. If the real part of a complex number is zero, then the number is called an [[imaginary number]] or is referred to as ''purely imaginary''; if the imaginary part is zero, then the number is a real number. Thus the real numbers are a [[subset]] of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called a [[Gaussian integer]]. The symbol for the complex numbers is '''C''' or <math>\mathbb{C}</math>.
In [[abstract algebra]], the complex numbers are an example of an [[algebraically closed field]], meaning that every [[polynomial]] with complex [[coefficient]]s can be [[Factorization|factored]] into linear factors. Like the real number system, the complex number system is a [[field (mathematics)|field]] and is [[completeness (order theory)|complete]], but unlike the real numbers it is not [[total order|ordered]]. That is, there is no meaning in saying that ''i'' is greater than 1, nor is there any meaning in saying that that ''i'' is less than 1. In technical terms, the complex numbers lack the [[trichotomy property]].
Complex numbers correspond to points on the [[complex plane]], sometimes called the Argand plane.
Each of the number systems mentioned above is a [[proper subset]] of the next number system. Symbolically, '''N''' ⊂ '''Z''' ⊂ '''Q''' ⊂ '''R''' ⊂ '''C'''.
===Computable numbers===
Moving to problems of computation, the '''[[computable number]]s''' are determined in the set of the real numbers. The computable numbers, also known as the '''recursive numbers''' or the '''computable reals''', are the [[real numbers]] that can be computed to within any desired precision by a finite, terminating [[algorithm]]. Equivalent definitions can be given using [[Mu-recursive function|μ-recursive function]]s, [[Turing machines]] or [[lambda-calculus|λ-calculus]] as the formal representation of algorithms. The computable numbers form a [[real closed field]] and can be used in the place of real numbers for many, but not all, mathematical purposes.
===Other types===
[[hyperreal number|Hyperreal]] and hypercomplex numbers are used in [[non-standard analysis]]. The hyperreals, or '''nonstandard reals''' (usually denoted as *'''R'''), denote an [[ordered field]] which is a proper [[Field extension|extension]] of the ordered field of [[real number]]s '''R''' and which satisfies the [[transfer principle]]. This principle allows true [[first-order logic|first order]] statements about '''R''' to be reinterpreted as true first order statements about *'''R'''.
[[Superreal number|Superreal]] and [[surreal number]]s extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form [[field (mathematics)|field]]s.
The idea behind [[p-adic number]]s is this:
While real numbers may have infinitely long expansions to the right of the decimal point, these numbers allow for infinitely long expansions to the left. The number system which results depends on what [[radix|base]] is used for the digits: any base is possible, but a system with the best mathematical properties is obtained when the base is a [[prime number]].
For dealing with infinite collections, the natural numbers have been generalized to the [[ordinal number]]s and to the [[cardinal number]]s. The former gives the ordering of the collection, while the latter gives its size. For the finite set, the ordinal and cardinal numbers are equivalent, but they differ in the infinite case.
There are also other sets of numbers with specialized uses. Some are subsets of the complex numbers. For example, [[algebraic numbers]] are the roots of [[polynomials]] with rational [[coefficients]]. Complex numbers that are not algebraic are called [[transcendental numbers]].
Sets of numbers that are not subsets of the complex numbers are sometimes called [[hypercomplex number]]s. They include the [[quaternion]]s '''H''', invented by Sir [[William Rowan Hamilton]], in which multiplication is not [[commutative]], and the [[octonion]]s, in which multiplication is not [[associative]]. Elements of [[Function field of an algebraic variety|function field]]s of non-zero [[characteristic (algebra)|characteristic]] behave in some ways like numbers and are often regarded as numbers by number theorists.
In addition, various specific kinds of numbers are studied in sets of [[natural number|natural]] and [[integer number]]s.
An '''even number''' is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an '''odd number''' is an integer that is not evenly divisible by 2. (The old-fashioned term "evenly divisible" is now almost always shortened to "[[Divisibility|divisible]]".)
A formal definition of an odd number is that it is an integer of the form ''n'' = 2''k'' + 1, where ''k'' is an integer. An even number has the form ''n'' = 2''k'' where ''k'' is an [[integer]].
A '''perfect number''' is defined as a [[Negative and non-negative numbers|positive integer]] which is the sum of its proper positive [[divisor]]s, that is, the sum of the positive divisors not including the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, or [[divisor function|σ]](''n'') = 2 ''n''. The first perfect number is [[6 (number)|6]], because 1, 2, and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is [[28 (number)|28]] = 1 + 2 + 4 + 7 + 14. The next perfect numbers are [[496 (number)|496]] and [[8128 (number)|8128]] {{OEIS|id=A000396}}. These first four perfect numbers were the only ones known to early [[Greek mathematics]].
A '''figurate number''' is a number that can be represented as a regular and discrete [[geometry|geometric]] pattern (e.g. dots). If the pattern is [[polytope|polytopic]], the figurate is labeled a '''polytopic number''', and may be a '''[[polygonal number]]''' or a '''polyhedral number'''. Polytopic numbers for r = 2, 3, and 4 are:
*''P''<sub>2</sub>(n) = 1/2 ''n''(''n'' + 1) ([[triangular number]]s)
*''P''<sub>3</sub>(n) = 1/6 ''n''(''n'' + 1)(''n'' + 2) ([[tetrahedral number]]s)
*''P''<sub>4</sub>(n) = 1/24 ''n''(''n'' + 1)(''n'' + 2)(''n'' + 3) ([[pentatopic number]]s)
==Numerals==
Numbers should be distinguished from ''[[Number names|numeral]]s'', the symbols used to represent numbers. The number five can be represented by both the base ten numeral '5' and by the [[Roman numeral]] 'V'. Notations used to represent numbers are discussed in the article [[numeral system]]s. An important development in the history of numerals was the development of a positional system, like modern decimals, which can represent very large numbers. The Roman numerals require extra symbols for larger numbers.
==History==
===History of integers===
==== The first use of numbers ====
It is speculated that the first known use of numbers dates back to around 30000 BC, bones or other artifacts have been discovered with marks cut into them which are often considered [[tally marks]]. The use of these tally marks have been suggested to be anything from counting elapsed time, such as numbers of days, or keeping records of amounts.
Tallying systems have no concept of place-value (such as in the currently used decimal notation), which limit its representation of large numbers and as such is often considered that this is the first kind of abstract system that would be used, and could be considered a Numeral System.
The first known system with place-value was the [[Ancient Mesopotamian units of measurement|Mesopotamian]] base 60 system (ca. 3400 BC) and the earliest known base 10 system dates to 3100 BC in [[Egypt]]. [http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin]
==== History of zero ====
{{further|[[0 (number)#History|History of zero]]}}
The use of zero as a number should be distinguished from its use as a placeholder numeral in [[place-value system]]s. Many ancient Indian texts use a [[Sanskrit]] word ''Shunya'' to refer to the concept of ''void''; in mathematics texts this word would often be used to refer to the number zero. [http://sunsite.utk.edu/math_archives/.http/hypermail/historia/apr99/0197.html]. In a similar vein, [[Pāṇini]] (5th century BC) used the null (zero) operator (ie a [[lambda production]]) in the [[Ashtadhyayi]], his [[formal grammar|algebraic grammar]] for the [[Sanskrit]] language. (also see [[Pingala]])
Records show that the [[Ancient Greece|Ancient Greeks]] seemed unsure about the status of zero as a number: they asked themselves "how can 'nothing' be something?" leading to interesting [[philosophy|philosophical]] and, by the Medieval period, religious arguments about the nature and existence of zero and the [[vacuum]]. The [[Zeno's paradoxes|paradoxes]] of [[Zeno of Elea]] depend in large part on the uncertain interpretation of zero. (The ancient Greeks even questioned if [[1 (number)|1]] was a number.)
The late [[Olmec]] people of south-central [[Mexico]] began to use a true zero (a shell glyph) in the New World possibly by the 4th century BC but certainly by 40 BC, which became an integral part of [[Maya numerals]] and the [[Maya calendar]], but did not influence Old World numeral systems.
By 130, [[Ptolemy]], influenced by [[Hipparchus]] and the Babylonians, was using a symbol for zero (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic [[Greek numerals]]. Because it was used alone, not as just a placeholder, this [[Greek numerals#Hellenistic zero|Hellenistic zero]] was the first ''documented'' use of a true zero in the Old World. In later [[Byzantine Empire|Byzantine]] manuscripts of his ''Syntaxis Mathematica'' (''Almagest''), the Hellenistic zero had morphed into the [[Greek alphabet|Greek letter]] [[omicron]] (otherwise meaning 70).
Another true zero was used in tables alongside [[Roman numerals#zero|Roman numerals]] by 525 (first known use by [[Dionysius Exiguus]]), but as a word, ''nulla'' meaning ''nothing'', not as a symbol. When division produced zero as a remainder, ''nihil'', also meaning ''nothing'', was used. These medieval zeros were used by all future medieval [[computus|computists]] (calculators of [[Easter]]). An isolated use of their initial, N, was used in a table of Roman numerals by [[Bede]] or a colleague about 725, a true zero symbol.
An early documented use of the zero by [[Brahmagupta]] (in the [[Brahmasphutasiddhanta]]) dates to 628. He treated zero as a number and discussed operations involving it, including [[division by zero|division]]. By this time (7th century) the concept had clearly reached [[Cambodia]], and documentation shows the idea later spreading to [[China]] and the [[Islam]]ic world.
==== History of negative numbers ====
{{further|[[Negative and non-negative numbers#First usage of negative numbers|First usage of negative numbers]]}}
The abstract concept of negative numbers was recognised as early as 100 BC - 50 BC. The [[China|Chinese]] ''”[[Nine Chapters on the Mathematical Art]]”'' (''Jiu-zhang Suanshu'') contains methods for finding the areas of figures; red rods were used to denote positive [[coefficient]]s, black for negative. This is the earliest known mention of negative numbers in the East; the first reference in a western work was in the 3rd century in [[Greece]]. [[Diophantus]] referred to the equation equivalent to <math>4x + 20 = 0</math> (the solution would be negative) in ''[[Arithmetica]]'', saying that the equation gave an absurd result.
During the 600s, negative numbers were in use in [[India]] to represent debts. [[Diophantus]]’ previous reference was discussed more explicitly by Indian mathematician [[Brahmagupta]], in [[Brahmasphutasiddhanta|Brahma-Sphuta-Siddhanta]] 628, who used negative numbers to produce the general form [[Quadratic equation#Quadratic formula|quadratic formula]] that remains in use today. However, in the 12th century in India, [[Bhaskara]] gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots."
[[Europe]]an mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although [[Leonardo of Pisa#Important publications|Fibonacci]] allowed negative solutions in financial problems where they could be interpreted as debits (chapter 13 of ''[[Liber Abaci]]'', 1202) and later as losses (in ''Flos''). At the same time, the [[China|Chinese]] were indicating negative numbers by drawing a diagonal stroke through the right-most nonzero digit of the corresponding positive number's numeral{{Fact|date=February 2007}}. The first use of negative numbers in a European work was by [[Chuquet]] during the 15th century. He used them as [[exponent]]s, but referred to them as “absurd numbers”.
As recently as the 18th century, the [[Switzerland|Swiss]] mathematician [[Leonhard Euler]] believed that negative numbers were greater than [[infinity]]{{Fact|date=February 2007}}, and it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless, just as [[Descartes|René Descartes]] did with negative solutions in a [[cartesian coordinate system]].
===History of rational, irrational, and real numbers===
{{further|[[irrational number#History|History of irrational numbers]] and [[History of pi]]}}
====History of rational numbers====
It is likely that the concept of fractional numbers dates to [[prehistoric times]]. Even the [[Ancient Egyptians]] wrote math texts describing how to convert general [[vulgar fraction|fractions]] into their [[Egyptian fractions|special notation]]. Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of [[number theory]]. The best known of these is [[Euclid's Elements]], dating to roughly 300 BC. Of the Indian texts, the most relevant is the [[Sthananga Sutra]], which also covers number theory as part of a general study of mathematics.
The concept of [[decimal fraction]]s is closely linked with decimal place value notation; the two seem to have developed in tandem. For example, it is common for the Jain math sutras to include calculations of decimal-fraction approximations to [[pi]] or the [[square root of two]]. Similarly, Babylonian math texts had always used sexagesimal fractions with great frequency.
====History of irrational numbers====
The earliest known use of irrational numbers was in the [[Indian mathematics|Indian]] [[Sulba Sutras]] composed between 800-500 BC. {{fact|date=April 2007}} The first existence proofs of irrational numbers is usually attributed to [[Pythagoras]], more specifically to the [[Pythagoreanism|Pythagorean]] [[Hippasus|Hippasus of Metapontum]], who produced a (most likely geometrical) proof of the irrationality of the [[square root of two|square root of 2]]. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However [[Pythagoras]] believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but his beliefs would not accept the existence of irrational numbers and so he sentenced Hippasus to death by drowning.
The sixteenth century saw the final acceptance by Europeans of [[negative and non-negative numbers|negative]], integral and [[fraction (mathematics)|fractional]] numbers. The seventeenth century saw decimal fractions with the modern notation quite generally used by mathematicians. But it was not until the nineteenth century that the irrationals were separated into algebraic and transcendental parts, and a scientific study of theory of irrationals was taken once more. It had remained almost dormant since [[Euclid]]. The year 1872 saw the publication of the theories of [[Karl Weierstrass]] (by his pupil [[Kossak]]), [[Eduard Heine|Heine]] (''[[Crelle]]'', 74), [[Georg Cantor]] (Annalen, 5), and [[Richard Dedekind]]. [[Méray]] had taken in 1869 the same point of departure as [[Eduard Heine|Heine]], but the theory is generally referred to the year 1872. Weierstrass's method has been completely set forth by [[Salvatore Pincherle]] (1880), and Dedekind's has received additional prominence through the author's later work (1888) and the recent endorsement by [[Paul Tannery]] (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a [[Dedekind cut|cut (Schnitt)]] in the system of [[real number]]s, separating all [[rational number]]s into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, [[Kronecker]] (Crelle, 101), and Méray.
[[Continued fraction]]s, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of [[Euler]], and at the opening of the nineteenth century were brought into prominence through the writings of [[Joseph Louis Lagrange]]. Other noteworthy contributions have been made by [[Druckenmüller]] (1837), [[Kunze]] (1857), [[Lemke]] (1870), and [[Günther]] (1872). [[Ramus]] (1855) first connected the subject with [[determinant]]s, resulting, with the subsequent contributions of Heine, [[August Ferdinand Möbius|Möbius]], and [[Günther]], in the theory of Kettenbruchdeterminanten. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.
====Transcendental numbers and reals====
The first results concerning transcendental numbers were [[Johann Heinrich Lambert|Lambert's]] 1761 proof that π cannot be rational, and also that ''e''<sup>''n''</sup> is irrational if ''n'' is rational (unless ''n'' = 0). (The constant [[E (mathematical constant)|''e'']] was first referred to in [[John Napier|Napier's]] 1618 work on [[logarithms]].) [[Legendre]] extended this proof to showed that π is not the square root of a rational number. The search for roots of [[Quintic equation|quintic]] and higher degree equations was an important development, the [[Abel–Ruffini theorem]] ([[Paolo Ruffini|Ruffini]] 1799, [[Niels Henrik Abel|Abel]] 1824) showed that they could not be solved by [[Radical (mathematics)|radicals]] (formula involving only arithmetical operations and roots). Hence it was necessary to consider the wider set of [[algebraic numbers]] (all solutions to polynomial equations). [[Évariste Galois|Galois]] (1832) linked polynomial equations to [[group theory]] giving rise to the field of [[Galois theory]].
Even the set of algebraic numbers was not sufficient and the full set of real number includes [[transcendental numbers]]. The existence of which was first established by [[Joseph Liouville|Liouville]] (1844, 1851). [[Charles Hermite|Hermite]] proved in 1873 that [[E (mathematical constant)|''e'']] is transcendental and [[Ferdinand von Lindemann|Lindemann]] proved in 1882 that π is transcendental. Finally [[Georg Cantor|Cantor]] shows that the set of all [[real number]]s is [[uncountable|uncountably infinite]] but the set of all [[algebraic number]]s is [[countable|countably infinite]], so there is an uncountably infinite number of transcendental numbers.
===Infinity===
{{further|[[Infinity#History|History of infinity]]}}
The earliest known conception of mathematical [[infinity]] appears in the [[Yajur Veda]] - an ancient script in India, which at one point states "if you remove a part from infinity or add a part to infinity, still what remains is infinity". Infinity was a popular topic of philosophical study among the [[Jain]] mathematicians circa 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.
In the West, the traditional notion of mathematical infinity was defined by [[Aristotle]], who distinguished between [[actual infinity]] and [[potential infinity]]; the general consensus being that only the latter had true value. [[Galileo]]'s [[Two New Sciences]] discussed the idea of [[bijection|one-to-one correspondences]] between infinite sets. But the next major advance in the theory was made by [[Georg Cantor]]; in 1895 he published a book about his new [[set theory]], introducing, among other things, [[transfinite number]]s and formulating the [[continuum hypothesis]]. This was the first mathematical model that represented infinity by numbers and gave rules for operating with these infinite numbers.
In the 1960s, [[Abraham Robinson]] showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. The system of '''hyperreal numbers''' represents a rigorous method of treating the ideas about [[infinite]] and [[infinitesimal]] numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of [[calculus]] by [[Isaac Newton|Newton]] and [[Gottfried Leibniz|Leibniz]].
A modern geometrical version of infinity is given by [[projective geometry]], which introduces "ideal points at infinity," one for each spatial direction. Each family of parallel lines in a given direction is postulated to converge to the corresponding ideal point. This is closely related to the idea of vanishing points in [[perspective (graphical)|perspective]] drawing.
===Complex numbers===
{{further|[[Complex number#History|History of complex numbers]]}}
The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor [[Heron of Alexandria]] in the 1st century AD, when he considered the volume of an impossible [[frustum]] of a [[pyramid]]. They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians (see [[Niccolo Fontana Tartaglia]], [[Gerolamo Cardano]]). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers.
This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. The term "imaginary" for these quantities was coined by [[René Descartes]] in 1637 and was meant to be derogatory (see [[imaginary number]] for a discussion of the "reality" of complex numbers). A further source of confusion was that the equation
:<math>\left ( \sqrt{-1}\right )^2 =\sqrt{-1}\sqrt{-1}=-1</math>
seemed to be capriciously inconsistent with the algebraic identity
:<math>\sqrt{a}\sqrt{b}=\sqrt{ab}</math>,
which is valid for positive real numbers ''a'' and ''b'', and which was also used in complex number calculations with one of ''a'', ''b'' positive and the other negative. The incorrect use of this identity, and the related identity
:<math>\frac{1}{\sqrt{a}}=\sqrt{\frac{1}{a}}</math>
in the case when both ''a'' and ''b'' are negative even bedeviled [[Euler]]. This difficulty eventually led him to the convention of using the special symbol ''i'' in place of √−1 to guard against this mistake.
The 18th century saw the labors of [[Abraham de Moivre]] and [[Leonhard Euler]]. To De Moivre is due (1730) the well-known formula which bears his name, [[de Moivre's formula]]:
:<math>(\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n \theta \,</math>
and to Euler (1748) [[Euler's formula]] of [[complex analysis]]:
:<math>\cos \theta + i\sin \theta = e ^{i\theta }. \,</math>
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. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in [[John Wallis|Wallis]]'s ''De Algebra tractatus''.
Also in 1799, Gauss provided the first generally accepted proof of the [[fundamental theorem of algebra]], showing that every polynomial over the complex numbers has a full set of solutions in that realm. The general acceptance of the theory of complex numbers is not a little due to the labors of [[Augustin Louis Cauchy]] and [[Niels Henrik Abel]], and especially the latter, who was the first to boldly use complex numbers with a success that is well known.
[[Carl Friedrich Gauss|Gauss]] studied [[Gaussian integer|complex numbers of the form]] ''a'' + ''bi'', where ''a'' and ''b'' are integral, or rational (and ''i'' is one of the two roots of ''x''<sup>2</sup> + 1 = 0). His student, [[Ferdinand Eisenstein]], studied the type ''a'' + ''bω'', where ''ω'' is a complex root of ''x''<sup>3</sup> − 1 = 0. Other such classes (called [[cyclotomic fields]]) of complex numbers are derived from the [[roots of unity]] ''x''<sup>''k''</sup> − 1 = 0 for higher values of ''k''. This generalization is largely due to [[Ernst Kummer]], who also invented [[ideal number]]s, which were expressed as geometrical entities by [[Felix Klein]] in 1893. The general theory of fields was created by [[Évariste Galois]], who studied the fields generated by the roots of any polynomial equation ''F''(''x'') = 0.
In 1850 [[Victor Alexandre Puiseux]] took the key step of distinguishing between poles and branch points, and introduced the concept of [[mathematical singularity|essential singular points]]; this would eventually lead to the concept of the [[extended complex plane]].
===Prime numbers===
[[Prime number]]s have been studied throughout recorded history. Euclid devoted one book of the ''Elements'' to the theory of primes; in it he proved the infinitude of the primes and the [[fundamental theorem of arithmetic]], and presented the [[Euclidean algorithm]] for finding the [[greatest common divisor]] of two numbers.
In 240 BC, [[Eratosthenes]] used the [[Sieve of Eratosthenes]] to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the [[Renaissance]] and later eras.
In 1796, [[Adrien-Marie Legendre]] conjectured the [[prime number theorem]], describing the asymptotic distribution of primes. Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the [[Goldbach conjecture]] which claims that any sufficiently large even number is the sum of two primes. Yet another conjecture related to the distribution of prime numbers is the [[Riemann hypothesis]], formulated by [[Bernhard Riemann]] in 1859. The prime number theorem was finally proved by [[Jacques Hadamard]] and [[Charles de la Vallée-Poussin]] in 1896. The conjectures of Goldbach and Riemann yet remain to be proved or refuted.
==References==
{{wiktionarypar|number}}
* [[Erich Friedman]], ''[http://www.stetson.edu/~efriedma/numbers.html What's special about this number?]''
* [[Steven Galovich]], ''Introduction to [[Mathematical structure|Mathematical Structure]]s'', Harcourt Brace Javanovich, [[23 January]] [[1989]], ISBN 0-15-543468-3.
* [[Paul Halmos]], ''Naive Set Theory'', Springer, 1974, ISBN 0-387-90092-6.
* [[Morris Kline]], ''Mathematical Thought from Ancient to Modern Times'', Oxford University Press, 1972.
* Whitehead and Russell, ''[[Principia Mathematica]]'' to *56, Cambridge University Press, 1910.
* [http://www.cut-the-knot.org/do_you_know/numbers.shtml What's a Number?] at [[cut-the-knot]]
==See also==
<div style="-moz-column-count:3; column-count:3;">
* [[Hebrew numerals]]
* [[Arabic numeral system]]
* [[Even and odd numbers]]
* [[Floating point|Floating point representation in computers]]
* [[Large numbers]]
* [[List of numbers]]
* [[List of numbers in various languages]]
* [[Mathematical constant]]s
* [[Mythical number]]s
* [[Negative and non-negative numbers]]
* [[Orders of magnitude]]
* [[Physical constant]]s
* [[Prime number]]s
* [[Small number]]s
* [[Subitizing and counting]]
* [[Number sign]]
* [[Numero sign]]
* [[0 (number)|Zero]]
* [[Pi]]
* [[The Foundations of Arithmetic]]
</div>
==External links==
{{Commons|Numbers}}
*[http://freepages.history.rootsweb.com/~catshaman/13comp/0numer.htm Mesopotamian and Germanic numbers]
*[http://www.bbc.co.uk/radio4/history/inourtime/inourtime_20060309.shtml BBC Radio 4, In Our Time: Negative Numbers]
*[http://www.gresham.ac.uk/event.asp?PageId=45&EventId=622 '4000 Years of Numbers'], lecture by Robin Wilson, 07/11/07, [[Gresham College]] (available for download as MP3 or MP4, and as a text file).
[[Category:Group theory]]
[[Category:Numbers| ]]
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