Number

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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 (highway and telephone numbers), as indicators of order (serial numbers), and as codes (ISBN numbers). In mathematics, the definition of a number has extended to include abstractions such as negative numbers, transcendental numbers, and <math>i</math>, or <math>\sqrt{-1}</math>.

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.

Contents

Types of numbers

Numbers can be classified into sets called number systems.

Natural numbers

The most familiar numbers are the natural numbers, which to some mean the non-negative integers {0, 1, 2, …} and to others mean the positive integers {1, 2, 3, …}. The positive integers are referred to as the 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 <math>\mathbb{N}.</math>

Integers

Negative numbers are numbers which are less than zero. They are used to indicate a number that is opposite to the corresponding positive number (the absolute value), but equal in magnitude. 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 <math>\mathbb{Z}</math> (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 1/2 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 <math>\mathbb{Q}</math> (for quotient).

Real numbers

Decimal numbers 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 (US) or a comma (Europe) to the right of the ones place. A rational number expressed as a decimal must terminate or repeat. An example of the former is 1.25; an example of the latter is 0.3333... (unending threes). Since they are rational, all repeating and terminating decimals can be written as fractions; using the examples above, 1.25 = 5/4 and 0.3333.... = 1/3. A repeating decimal represents an infinite series. Unlike repeating and terminating decimals, some decimals can not be written as fractions. Those include 0.1010010001... as well as constants such as π (pi), and <math>\sqrt{2}</math>, the square root of 2. A number that cannot be written as a fraction is called an irrational number.

The real numbers are made up of all numbers that can be expressed as decimals, both rational and irrational. The symbol for the real numbers is <math>\mathbb{R}.</math> The real numbers are used to represent measurements, and correspond to the points on the number line. In a technical sense, the real numbers are the ideal set of numbers. They are the only complete ordered field; however, they are not an algebraically closed field.

Complex numbers

Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers <math>\mathbb{C}.</math> 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, <math>\mathbb{N} \sub \mathbb{Z} \sub \mathbb{Q} \sub \mathbb{R} \sub \mathbb{C}.</math>

Infinitary extensions

Superreal, hyperreal and surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers. 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. This leads to the p-adic numbers.

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.

Other types

While the natural numbers or the real numbers suffice for most everyday purposes, mathematicians have discovered 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. Real numbers that are not algebraic are called transcendental numbers. Sets of numbers that are not subsets of the complex numbers include the quaternions <math>\mathbb{H}</math>, 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

See also history of numeral systems

The first known use of numbers dates back to around 30000 BC when tally marks were used by stone age peoples. The earliest known example is from a cave in Southern Africa. [1]. This system had no concept of base, which limited its representation of large numbers. The first known system with a base was the Mesopotamian base 60 system (ca. 3400 BC) and the earliest known base 10 system dates to 3100 BC in Egypt. [2]

History of zero

See also 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. [3]. 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

See also First usage 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 <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 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. 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.

History of rational, irrational, and real numbers

See also: history of irrational numbers, 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 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 en 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

See also: history of 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.

Complex numbers

See also: history of 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 <math>\sqrt{-1}^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 <math>\sqrt{-1}</math> 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 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 <math>a + bi</math>, where a and b are integral, or rational (and i is one of the two roots of <math>x^2 + 1 = 0</math>). His student, Ferdinand Eisenstein, studied the type <math>a + b\omega</math>, where <math>\omega</math> is a complex root of <math>x^3 - 1 = 0</math>. Other such classes (called cyclotomic fields) of complex numbers are derived from the roots of unity <math>x^k - 1 = 0</math> for higher values of <math>k</math>. 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

<math>\ F(x) = 0.</math>

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.

References


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See also

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