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A number is an abstract idea used in the counting and measurement. A symbol that represents a number is called a numeral, but in the common use of the word number is used both for the idea and the symbol. In addition to their use in counting and measurement, numbers are often used to label (phone numbers), in order (serial numbers), and by codes (ISBNs). In mathematics, defining the number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers and complex numbers. On the basis ten number system, in use today almost universal, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. On this basis ten system, the right digit of a natural number has a value of a place, and all other figures have a place ten times the value of the place value of the digit to the right. The symbol for the set of all natural numbers is N, also in writing. Negative numbers 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 in front of the number they are the opposite of. Thus, the opposite of 7 is written -7. When the number of negative integers are combined with the positive whole numbers and zero, one gets the whole Z (German Zahl, Zahlen plural), also in writing. If the absolute value of m is greater than n, then the absolute value of the fraction is greater than 1. Fractions can be higher, lower or equal to 1 and also can be positive, negative or zero. The set of all fractions includes the whole, since every integer can be written as a fraction with denominator 1. For example -7 can be written -7 / 1. The symbol for rational numbers is Q (per ratio), also in writing. The real numbers include all issues of measurement. Real numbers are usually written using decimal numbers in a decimal point is placed on the right of the digit with a value spot. After the decimal point, each digit is a place value one tenth of the value of the site for the left digit. Thus, represents 1 hundred, 2 dozens, 3 ones, 4 tenths, 5 hundredths, thousandths and 6. In saying the number, the decimal is read "point", thus: "one point two three four five six". In, for example, the E.U. And the United Kingdom, the decimal is represented by a period in continental Europe by a comma. Zero is often written as 0.0 and negative real numbers are written with the first sign: Move to a higher level of abstraction, the real numbers can be extended to complex numbers. This set of figures emerged, historically, from the question of whether a negative number can be a square root. This led to the invention of a new issue: the square root of a negative, denoted by i, a symbol assigned by Leonhard Euler, and called for the imaginary unit. The figures complex consisting of all issues in the way Where a and b are real numbers. In the words + bi, the actual number is called the real part and b is called the imaginary part. If the real part of a complex number is zero, then the number is called an imaginary or is mentioned as purely imaginary; is the imaginary part is zero, then the number is a real number. Thus, the real numbers are a subset of complex numbers. If the real and imaginary parts of a complex number are both whole, then the number is called Gaussian integer. The symbol for the complex numbers is C or. In abstract algebra, complex numbers are an example of a field algebraically closed, which means that every polynomial coefficients with complex can be integrated into linear factors. Like the real number system, the number system is a complex area and is complete, but unlike the real numbers, is not ordained. That is, there is no sense in what I say is more than 1, nor is there any meaning in saying that i is less than 1. In technical terms, the numbers lack the complex trichotomy property. The idea behind p-add numbers is this: While actual numbers may have infinitely long expansions to the right of the decimal point, these numbers allow the infinitely long expansions to the left. The number system, which depends on results that base is used for the digits: any basis is possible, but a system with the best mathematical properties is obtained when the base is a prime number. To deal with infinite collections, the natural numbers were generalized ordinal numbers and the numbers for the Cardinals. The former gave the ordering of the collection, while the second gave his size. For all finite, the cardinal and ordinal numbers are equivalent, but differ in the case infinity. Sets of numbers that are not subsets of the complex numbers include quaternions H, invented by Sir William Rowan Hamilton, which is noncommutative multiplication, and octonions, in which multiplication is not associative. Elements of the function characteristic of finite fields behave, in some respects, such as numbers and are often regarded as serial numbers theoretical. Numbers must be distinguished from numbers, the symbols used to represent numbers. The number five can be represented by both the base in December numeral'5 'and the Roman numeral' V '. Ratings used to represent numbers are discussed in Article numeral systems. An important development in the history of numbering has been the development of a positional system, as the modern decimal, which can represent a large number. The Roman numerals require extra symbols for larger numbers. It is speculated that the first use of numbers known dates back to around 30000 BC, bones or other artifacts were discovered with cut marks where they are often considered registration marks. The use of these marks record were suggested to be something of counting time, such as numbers of days, or keep records of the amounts. Tallying systems have no concept of the place of value (as currently used in decimal notation), which limit its representation of a large number and, as such, is often considered that this is the first kind of abstract system that would be used, and could be Considered a system numbers. The use of zero as the number should be distinguished from its use as a placeholder numeral instead of value systems. Many ancient Indian texts using a Sanskrit word Shunya to refer to the concept of invalidity; in mathematics texts that word would often be used to refer to the number zero. [2]. Similarly, Pāṇini (5 th century BC) used the zero (zero) operator (ie a production lambda) in the Ashtadhyayi, algebraic their grammar for the language Sanskrit. (See also Pingala) records show that the ancient Greeks seemed unsure about the status of zero as a number: they asked themselves "how can" nothing "is a thing?" Taking interesting and philosophical, the medieval period, religious arguments about the nature and the existence of zero in the vacuum. The paradoxes of Zeno of Elea depend in large part on the interpretation of zero uncertain. (The ancient Greeks until 1 has been questioned whether a paragraph.) The late Olmec people of south-central Mexico began using a true zero (a shell glyph) in the New World possibly through the 4 th century BC, but certainly in 40 BC, which became an integral part of Maya numerals and the Maya calendar, but not influenced 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 system sexagesimal numeral one using alphabetical Greek numerals. Because it was used alone, not as just a space, this Hellenistic zero was the first documented use of a real zero in the Old World. In later Byzantine manuscripts of his Syntaxis Mathematica (Almagest), the Hellenistic zero had morphed the Greek letter omicron (another meaning 70). Another true zero was used in tables alongside Roman numerals by 525 (first use known as 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. One of the first documented use of zero by Brahmagupta (Brahmasphutasiddhanta) dates to 628. He treated zero as a number and discussed operations involving including division. At this time (7th century), the concept had clearly reached Cambodia, and documentation shows the idea later spreading to China and the Islamic world. During the 600s, negative numbers were in use in India to represent debts. Diophantus' reference has been discussed more explicitly by Indian mathematician Brahmagupta, in the Brahma-Sphuta-Siddhanta 628, which used to produce the negative numbers in general quadratic formula that remains in use today. However, the century 12 in India, Bhaskara gives negative roots of quadratic equations, but says that the negative value "is, in this case, not to be taken as it is inadequate; people do not approve of negative roots." It is likely that the concept of fractional numbers dates from pre-historic times. Even the ancient Egyptians wrote math texts describing how to convert fractions in particular its overall rating. Classic Greek and Indian studies made of the mathematical theory of rational numbers, as part of the global study of number theory. The best known is Euclid's Elements, which dates from about 300 BC. Do 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 related to the decimal value notation, the two seem to have developed in tandem. For example, it is common for mathematics Jain sutras to include calculations of approximations a decimal fraction-pi or the square root of two. Similarly, Babylonian mathematics texts had always used sexagesimal fractions with great frequency. In the sixteenth century, the acceptance by Europeans of negative final, integral and fractional numbers. XVII Century saw decimal fractions with the modern notation quite often used by mathematicians. But it was not until the nineteenth century that the irrationals have been split into algebraic and transcendental parts, and a scientific study of the theory of irrationals was taken once more. He had remained almost dormant since Euclid. The year 1872 saw the publication of the theories of Karl Weierstrass theorems (for 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 refers to the year 1872. Weierstrass theorems of the method has been completely defined by Salvatore Pincherle (1880), and Dedekind's received more prominence through the work of the author, later (1888) and the recent approval by Paul Tannery (1894). Weierstrass theorems, Cantor, and Heine base their theories about infinite series, while his Dedekind found in the idea of a cut (Schnitt) in the system of real numbers, separating all rational numbers into two groups with some characteristic properties. The subject has received contributions later at the hands of Weierstrass theorems, Kronecker (Crelle, 101) and Méray. Fractions continuous, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and the opening of the nineteenth century were brought to prominence through the writings of Joseph Louis Lagrange. Other notable contributions were made by Druckenmüller (1837), Kunze (1857), Lemke (1870), and Günther (1872). Ramus (1855) first connected with the theme determinants resulting, with the subsequent contributions of Heine, Möbius, and Günther, in the theory of Kettenbruchdeterminanten. Dirichlet also added to the general theory, as well as the many contributors to the applications of the subject. The first results relating transcendental numbers were Lambert's 1761 proves that π may not be rational, and that is irrational in if n is rational (unless n = 0). (A constant and was first mentioned in 1618 Napier's work on logarithms.) Legendre extended to this evidence showed that π is the square root of a rational number. The search for roots of quintic and greater degree equations has been an important development, the Abel-Ruffini theorem (Ruffini 1799, Abel 1824) showed that they could not be solved by radical (formula involving only the arithmetic operations and roots). Therefore, it was necessary to consider the wider set of algebraic numbers (all solutions of polynomial equations). Galois (1832) linked to the group polynomial equations theory that gave rise to the field of Galois theory. Even the set of algebraic numbers was not sufficient and complete the set of real numbers figure includes transcendental. The existence of which was first established by Liouville (1844, 1851). Hermite proved in 1873 and is transcendental and Lindemann proved in 1882 that π is transcendental. Finally Cantor shows that the set of all real numbers are uncountably infinite, but the set of all numbers is algebraic countably infinite, so there is an infinite number of uncountably transcendental numbers. The oldest known design of mathematical infinity appears in the Yajur Veda, which at one point states "if you remove a piece of infinity or add a part to infinity, what remains is infinite." Infinity has been a popular topic of study philosophical between the Jain mathematicians circa 400 BC. They distinguish between five types of infinity: infinite in one and two ways, in the infinite space, infinite everywhere, and infinite perpetually. In the West, the traditional concept of mathematical infinity was defined by Aristotle, which distinguish between real and infinite potential infinity, the general consensus that only the latter had real value. Galileo's Two New Sciences discussed the idea of one-to-one correspondence between sets infinite. But the next great breakthrough in the theory was done by Georg Cantor, in 1895 he published a book on his new set theory, introducing, among other things, the continuum hypothesis. A modern version of infinite geometrical is amended by projective geometry, which introduces "ideal points at infinity," one for each direction space. Each family of parallel lines in a certain direction is postulated to converge to the point corresponding ideal. This is closely related to the idea of escape points drawing in perspective. The first fleeting reference to the square roots of negative numbers occurred in the work of mathematician and inventor Heron of Alexandria in the 1 st century AD, when it considered the volume of an impossible frustum of a pyramid. They became more prominent when No 16 century closed formulas for the roots of the third and fourth degree polynomials were discovered by mathematical Italians (see Fontana Niccolo Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even when only one was interested in real solutions, sometimes required the manipulation of the square roots of negative numbers. This was doubly disturbing, since not even considered to be negative numbers on firm ground at the moment. The term "imaginary" for these quantities was coined by René Descartes by 1637 and was designed to be derogatory (see imaginary number for a discussion on the "reality" of complex numbers). Another source of confusion was that the equation the existence of complex numbers was not fully accepted until the geometrical interpretation had been described by Caspar Wessel in 1799, was rediscovered many years later and popularized by Carl Friedrich Gauss, and as a result the theory of Complex Numbers received a remarkable expansion. The idea of the graphical representation of complex numbers had appeared, however, once in 1685, in Wallis's De Algebra tractatus. Also in 1799, Gauss from the first generally accepted proof of the fundamental theorem of algebra, showing that every polynomial on the complex numbers has a complete set of solutions in this area. The general acceptance of the theory of complex numbers is not a little due to the work of Augustin Louis Cauchy and Niels Henrik Abel, and especially the latter, which was the first to use bravely complex numbers with a success that is well known. Gauss studied complex numbers of the form a + bi, where a and b are full, or rational (ei is one of the two roots of x2 + 1 = 0). His student, Ferdinand Eisenstein, studied the type ω a + b, where ω is a complex from scratch x3 - 1 = 0. Other these classes (called cyclotomic fields), complex numbers are derived from the root of the unit xk - 1 = 0 for higher values of k. This generalization is largely due to Ernst Kummer, who also invented ideal figures, which were expressed as geometric entities by Felix Klein, in 1893. The general theory of the fields was created by Évariste Galois, who studied the fields generated by the roots of any polynomial equation F (x) = 0. Prime numbers have been studied throughout recorded history. Euclides Elements of a book devoted to the theory of cousins, in which he revealed the infinitude of prime and fundamental theorem of arithmetic, and presented the Euclidean algorithm for finding the greatest common divisor of two numbers. In 1796, Adrien-Marie Legendre conjectured the prime number theorem, describing the asymptotic distribution of cousins. Other results concerning the distribution of cousins include Euler's proof that the sum of the reciprocals of cousins diverges, and the Goldbach conjecture, which says that any number large enough that 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 proven by Jacques Hadamard and Charles de la-Vallée Poussin in 1896.