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Number

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   A number is an abstract entity that represents a count or measurement.
   A symbol for a number is called a numeral. In common usage, numerals
   are often used as labels (e.g. road, telephone and house numbering), as
   indicators of order ( serial numbers), and as codes ( ISBN). In
   mathematics, the definition of a number has extended to include
   abstractions such as fractions, negative, irrational, transcendental
   and complex numbers.

   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.

Types of numbers

   Numbers can be classified into sets called number systems. For
   different methods of expressing numbers with symbols, see numeral
   systems.

Natural numbers

   The most familiar numbers are the natural numbers, which to some mean
   the non-negative integers and to others mean the positive integers. In
   everyday parlance the non-negative integers are commonly referred to as
   whole numbers, the positive integers as counting numbers.

   In the base ten number system, in almost universal use, the symbols for
   natural numbers are written using ten digits, 0 through 9. An implied
   place value system, one that increments in powers of ten, is used for
   numbers greater than nine. Thus, numbers greater than nine have
   numerals formed with two or more digits. The symbol for the set of all
   natural numbers is \mathbb{N}.

Integers

   Negative numbers are numbers which are less than zero. They are usually
   written by indicating their opposite, which is a positive number, with
   a preceding minus sign. For example, if a positive number is used to
   indicate distance to the right of a fixed point, a negative number
   would indicate distance to the left. Similarly, if a positive number
   indicates a bank deposit, then a negative number indicates a
   withdrawal. When the negative whole numbers are combined with the
   positive whole numbers and zero, one obtains the integers \mathbb{Z}
   (German Zahl, plural Zahlen).

Rational numbers

   A rational number is a number that can be expressed as a fraction with
   an integer numerator and a non-zero natural number denominator. The
   fraction m/n represents the quantity arrived at when a whole is divided
   into n equal parts, and m of those equal parts are chosen. Two
   different fractions may correspond to the same rational number; for
   example ½ and 2/4 are the same. If the absolute value of m is greater
   than n, the absolute value of the fraction is greater than one.
   Fractions can be positive, negative, or zero. The set of all fractions
   includes the integers, since every integer can be written as a fraction
   with denominator 1. The symbol for the rational numbers is a bold face
   \mathbb{Q} (for quotient).

Real numbers

   Loosely speaking, real numbers may be identified with points on a
   continuous line. All rational numbers are real numbers, and like those,
   real numbers can be classified as being either positive, zero, or
   negative.

   The real numbers are uniquely characterized by their mathematical
   properties: they are the only complete ordered field. They are not,
   however, an algebraically closed field.

   Decimal numerals are another way in which numbers can be expressed. In
   the base-ten number system, they are written as a string of digits,
   with a period ( decimal point) (in, for example, the US and UK) or a
   comma (in, for example, continental Europe) to the right of the ones
   place; negative real numbers are written with a preceding minus sign. A
   decimal numeral defining a rational number repeats or terminates
   (though any number of zeroes can be appended), though 0 is the only
   real number that cannot be defined through a repeating decimal numeral.
   For example, the fraction 5/4 can be written as the decimal numeral
   1.25, which terminates, or the decimal numeral 1.24999... (unending
   nines), which repeats. The fraction 1/3 can be written only as
   0.3333... (unending threes), which repeats. All repeating and
   terminating decimal numerals define rational numbers, which can also be
   written as fractions; 1.25 = 5/4 and 0.3333... = 1/3. Unlike repeating
   and terminating decimal numerals, non-repeating, non-terminating
   decimal numerals represent irrational numbers, numbers that cannot be
   written as fractions. For example, well-known mathematical constants
   such as π (pi) and \sqrt{2} , the square root of 2, are irrational, and
   so is the real number expressed by the decimal numeral
   0.101001000100001..., because that expression doesn't repeat or
   terminate.

   The real numbers are made up of all numbers that can be expressed as
   decimal numerals, both rational and irrational. The symbol for the real
   numbers is \mathbb{R}. The real numbers are used to represent
   measurements, and correspond to the points on the number line. As
   measurements are only made to a certain level of precision, there is
   always some error margin when using real numbers to represent them.
   This is often dealt with by giving an appropriate number of significant
   figures.

Complex numbers

   Moving to a greater level of abstraction, the real numbers can be
   extended to the complex numbers \mathbb{C}. This set of numbers arose,
   historically, from the question of whether a negative number can have a
   square root. From this problem, a new number was discovered: the square
   root of negative one. This number is denoted by i, a symbol assigned by
   Leonhard Euler. The complex numbers consist of all numbers of the form
   a + b i, where a and b are real numbers. When a is zero, then a + b i
   is called imaginary. Likewise, when b is zero, then a + b i is real,
   since there is no imaginary component. A complex number that has a and
   b as integers is called a Gaussian integer. The complex numbers are an
   algebraically closed field, meaning that every polynomial with complex
   coefficients can be factored into linear factors with complex
   coefficients. Complex numbers correspond to points on the complex
   plane.

   Each of the number systems mentioned above is a subset of the next
   number system. Symbolically, \mathbb{N} \sub \mathbb{Z} \sub \mathbb{Q}
   \sub \mathbb{R} \sub \mathbb{C}.

Other types

   Superreal, hyperreal and surreal numbers extend the real numbers by
   adding infinitesimally small numbers and infinitely large numbers, but
   still form fields.

   The idea behind p-adic numbers 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 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 numbers and to the cardinal numbers. 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 include the
   quaternions \mathbb{H} , invented by Sir William Rowan Hamilton, in
   which multiplication is not commutative, and the octonions, in which
   multiplication is not associative. Elements of function fields of
   finite characteristic behave in some ways like numbers and are often
   regarded as numbers by number theorists.

Numerals

   Numbers should be distinguished from numerals, 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 systems. 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 numbers

   The first known use of numbers dates back to around 30000 BC when tally
   marks were used by Paleolithic peoples. The earliest known example is
   from a cave in Southern Africa. . This system had no concept of
   place-value (such as in the currently used decimal notation), which
   limited its representation of large numbers. The first known system
   with place-value was the Mesopotamian base 60 system (ca. 3400 BC) and
   the earliest known base 10 system dates to 3100 BC in Egypt.

History of zero

   The use of zero as a number should be distinguished from its use as a
   placeholder numeral in place-value systems. 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. . In a similar vein, Pāṇini ( 5th century BC) used the null
   (zero) operator (ie a lambda production) in the Ashtadhyayi, his
   algebraic grammar for the Sanskrit language. (also see Pingala)

   Records show that the Ancient Greeks seemed unsure about the status of
   zero as a number: they asked themselves "how can 'nothing' be
   something?", leading to interesting philosophical and, by the Medieval
   period, religious arguments about the nature and existence of zero and
   the vacuum. The paradoxes of Zeno of Elea depend in large part on the
   uncertain interpretation of zero. (The ancient Greeks even questioned
   that 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 Hellenistic
   zero was the first documented use of a true zero in the Old World. In
   later Byzantine manuscripts of his Syntaxis Mathematica (Almagest), the
   Hellenistic zero had morphed into the Greek letter omicron (otherwise
   meaning 70).

   Another true zero was used in tables alongside 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 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 this time
   (7th century) the concept had clearly reached Cambodia, and
   documentation shows the idea later spreading to China and the Islamic
   world.

History of negative numbers

   The abstract concept of negative numbers was recognised as early as 100
   BC - 50 BC. The 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 coefficients, 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 4x + 20 = 0 (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 Brahma-Sphuta-Siddhanta 628, who
   used negative numbers to produce the general form 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."

   European mathematicians, for the most part, resisted the concept of
   negative numbers until the 17th century, although 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 Chinese were indicating
   negative numbers by drawing a diagonal stroke through the right-most
   nonzero digit of the corresponding positive number's numeral. The first
   use of negative numbers in a European work was by Chuquet during the
   15th century. He used them as exponents, but referred to them as
   “absurd numbers”.

   As recently as the 18th century, the Swiss mathematician Leonhard Euler
   believed that negative numbers were greater than infinity, and it was
   common practice to ignore any negative results returned by equations on
   the assumption that they were meaningless, just as René Descartes did
   with negative solutions in a cartesian coordinate system.

History of rational, irrational, and real numbers

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 fractions into their 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 fractions 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 Sulba
   Sutras composed between 800- 500 BC. The first existence proofs of
   irrational numbers is usually attributed to Pythagoras, more
   specifically to the Pythagorean Hippasus of Metapontum, who produced a
   (most likely geometrical) proof of the irrationality of the 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, integral and 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), Heine (
   Crelle, 74), Georg Cantor (Annalen, 5), and Richard Dedekind. Méray had
   taken in 1869 the same point of departure as Heine, but the theory is
   generally referred to the year 1872. Weierstrass's method has been
   completely set forth by 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 cut (Schnitt) in the system of real numbers,
   separating all rational numbers into two groups having certain
   characteristic properties. The subject has received later contributions
   at the hands of Weierstrass, Kronecker (Crelle, 101), and Méray.

   Continued fractions, 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
   determinants, resulting, with the subsequent contributions of Heine,
   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 Lambert's 1761
   proof that π cannot be rational, and also that e^n is irrational if n
   is rational (unless n = 0). (The constant e was first referred to in
   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 and higher degree equations was an important
   development, the Abel–Ruffini theorem ( Ruffini 1799, Abel 1824) showed
   that they could not be solved by 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). 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 Liouville (1844, 1851). Hermite proved in 1873
   that e is transcendental and Lindemann proved in 1882 that π is
   transcendental. Finally Cantor shows that the set of all real numbers
   is uncountably infinite but the set of all algebraic numbers is
   countably infinite, so there is an uncountably infinite number of
   transcendental numbers.

Infinity

   The earliest known conception of mathematical infinity appears in the
   Yajur Veda, 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
   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, the
   continuum hypothesis.

   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 drawing.

Complex numbers

   The earliest fleeting reference to square roots of negative numbers
   occurred in the work of the Greek 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 \sqrt{-1}^2=\sqrt{-1}\sqrt{-1}=-1 seemed to be capriciously
   inconsistent with the algebraic identity \sqrt{a}\sqrt{b}=\sqrt{ab} ,
   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 \frac{1}{\sqrt{a}}=\sqrt{\frac{1}{a}} ) 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
   \sqrt{-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:

          (\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n
          \theta \,

   and to Euler (1748) Euler's formula of complex analysis:

          \cos \theta + i\sin \theta = e ^{i\theta }. \,

   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 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.

   Gauss studied 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^2 + 1 = 0).
   His student, Ferdinand Eisenstein, studied the type a + bω, where ω is
   a complex root of x^3 − 1 = 0. Other such classes (called cyclotomic
   fields) of complex numbers are derived from the roots of unity x^k − 1
   = 0 for higher values of k. This generalization is largely due to
   Kummer, who also invented ideal numbers, 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
   essential singular points; this would eventually lead to the concept of
   the extended complex plane.

Prime numbers

   Prime numbers 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.
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