A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers.

Mathematical operations are certain procedures that take one or more numbers as input and produce a number as output. Unary operations take a single input number and produce a single output number. For example, the successor operation adds one to an integer, thus the successor of 4 is 5. Binary operations take two input numbers and produce a single output number. Examples of binary operations include addition, subtraction, multiplication, division, and exponentiation. The study of numerical operations is called arithmetic.

A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (telephone numbers), for ordering (serial numbers), and for codes (e.g., ISBNs).

In common use, the word number can mean the abstract object, the symbol, or the word for the number.

Classification of numbers[link]

Different types of numbers are used in many cases. Numbers can be classified into sets, called number systems. (For different methods of expressing numbers with symbols, such as the Roman numerals, see numeral systems.)

Natural numbers[link]

The most familiar numbers are the natural numbers or counting numbers: one, two, three, and so on. Traditionally, the sequence of natural numbers started with 1 (0 was not even considered a number for the Ancient Greeks.) However, in the 19th century, set theorists and other mathematicians started including 0 (cardinality of the empty set, i.e. 0 elements, where 0 is thus the smallest cardinal number) in the set of natural numbers.^{[citation needed]} Today, different mathematicians use the term to describe both sets, including zero or not. The mathematical symbol for the set of all natural numbers is N, also written Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{N} .

In the base ten numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten digits: 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.

In set theory, which is capable of acting as an axiomatic foundation for modern mathematics,^[1] 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 (i.e., 3 is the third successor of 0). Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols three times.

Integers[link]

The negative of a positive integer is defined as a number that produces zero when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign (a minus sign). As an example, the negative of 7 is written −7, and 7 + (−7) = 0. When the set of negative numbers is combined with the set of natural numbers (which includes zero), the result is defined as the set of integer numbers, also called integers, Z also written Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z} . Here the letter Z comes from German Zahl, meaning "number".

The set of integers forms a ring with operations addition and multiplication.^[2]

Rational numbers[link]

A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator. Fractions are written as two numbers, the numerator and the denominator, with a dividing bar between them. In the fraction written ^m⁄_n or

Failed to parse (Missing texvc executable; please see math/README to configure.): m \over n \,

m represents 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): {1 \over 2} = {2 \over 4}.\,

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Q} .

Real numbers[link]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): 123.456\,

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 period, whereas in continental Europe and certain other countries the decimal point is represented by a comma. Zero is often written as 0.0 when it must be treated as a real number rather than an integer. In the US and UK a number between −1 and 1 is always written with a leading zero to emphasize the decimal. Negative real numbers are written with a preceding minus sign:

Failed to parse (Missing texvc executable; please see math/README to configure.): -123.456.\,

Every rational number is also a real number. 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. 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, otherwise written 0.3) 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \pi = 3.14159265358979\dots.\,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{2} = 1.41421356237 \dots\,

(the square root of 2, that is, the positive number whose square is 2).

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, also written as Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R} .

When a real number represents a measurement, there is always a margin of error. This is often indicated by rounding or 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 0.001 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, it can be shown that any complete ordered field is isomorphic to the real numbers. The real numbers are not, however, an algebraically closed field.

Complex numbers[link]

Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. This set of numbers arose, historically, from trying to find closed formulas for the roots of cubic and quartic polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: the square root of negative one, denoted by i, a symbol assigned by Leonhard Euler, and called the imaginary unit. The complex numbers consist of all numbers of the form

Failed to parse (Missing texvc executable; please see math/README to configure.): \,a + b i

where a and b are real numbers. In the expression a + bi, the real number a 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 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{C} .

In abstract algebra, the complex numbers are an example of an algebraically closed field, meaning that every polynomial with complex coefficients can be factored into linear factors. Like the real number system, the complex number system is a field and is complete, but unlike the real numbers it is not ordered. That is, there is no meaning in saying that i is greater than 1, nor is there any meaning in saying 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, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} .

Computable numbers[link]

Moving to problems of computation, the computable numbers 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 μ-recursive functions, Turing machines or 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[link]

Algebraic numbers are those that can be expressed as the solution to a polynomial equation with integer coefficients. The complement of the algebraic numbers are the transcendental numbers.

Hyperreal numbers are used in non-standard analysis. The hyperreals, or nonstandard reals (usually denoted as *R), denote an ordered field that is a proper extension of the ordered field of real numbers R and satisfies the transfer principle. This principle allows true first order statements about R to be reinterpreted as true first order statements about *R.

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

The p-adic numbers may have infinitely long expansions to the left of the decimal point, in the same way that real numbers may have infinitely long expansions to the right. The number system that results depends on what base is used for the digits: any base is possible, but a prime number base provides the best mathematical properties.

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.

A relation number is defined as the class of relations consisting of all those relations that are similar to one member of the class.^[3]

Sets of numbers that are not subsets of the complex numbers are sometimes called hypercomplex numbers. They include the quaternions 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 non-zero characteristic behave in some ways like numbers and are often regarded as numbers by number theorists.

Specific uses[link]

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.

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 "divisible".) A formal definition of an odd number is that it is an integer of the form n = 2k + 1, where k is an integer. An even number has the form n = 2k where k is an integer.

A perfect number is a positive integer that is the sum of its proper positive divisors—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 σ(n) = 2 n. The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128 (sequence A000396 in OEIS). 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 geometric pattern (e.g. dots). If the pattern is 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:

• P₂(n) = ¹⁄₂ n(n + 1) (triangular numbers)
• P₃(n) = ¹⁄₆ n(n + 1)(n + 2) (tetrahedral numbers)
• P₄(n) = ¹⁄₂₄ n(n + 1)(n + 2)(n + 3) (pentatopic numbers)

Numerals[link]

Numbers should be distinguished from numerals, the symbols used to represent numbers. Boyer showed that Egyptians created the first ciphered numeral system.^{[citation needed]} Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. The number five can be represented by both the base ten numeral '5', by the Roman numeral 'Ⅴ' and ciphered letters. 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[link]

First use of numbers[link]

Bones and other artifacts have been discovered with marks cut into them that many believe are tally marks.^[4] These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals.

A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers. Nonetheless tallying systems are considered the first kind of abstract numeral system.

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.^[5]

[edit] Zero

The use of zero as a number should be distinguished from its use as a placeholder numeral in place-value systems. Many ancient texts used zero. Babylonian and Egyptian texts used it. Egyptians used the word nfr to denote zero balance in double entry accounting entries. Indian texts used a Sanskrit word Shunye to refer to the concept of void. In mathematics texts this word often refers to the number zero.^[6]

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 whether 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. Mayan arithmetic used base 4 and base 5 written as base 20. Sanchez in 1961 reported a base 4, base 5 'finger' abacus.

By 130 AD, 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 (the 7th century) the concept had clearly reached Cambodia as Khmer numerals, and documentation shows the idea later spreading to China and the Islamic world.

[edit] 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 (Chinese: Jiu-zhang Suanshu) contains methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative.^[7] 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 Failed to parse (Missing texvc executable; please see math/README to configure.): 4x + 20 = 0

(the solution is 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 debts (chapter 13 of Liber Abaci, 1202) and later as losses (in Flos). At the same time, the Chinese were indicating negative numbers either by drawing a diagonal stroke through the right-most nonzero digit of the corresponding positive number's numeral.^[8] 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, 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.

[edit] Rational numbers

It is likely that the concept of fractional numbers dates to prehistoric times. The Ancient Egyptians used their Egyptian fraction notation for rational numbers in mathematical texts such as the Rhind Mathematical Papyrus and the Kahun Papyrus. 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 (base 60) fractions with great frequency.

[edit] Irrational numbers

The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800–500 BC.^[9] 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 he could not accept irrational numbers, so he sentenced Hippasus to death by drowning.

The sixteenth century brought final European acceptance of negative integral and fractional numbers. By the seventeenth century, mathematicians generally used decimal fractions with modern notation. It was not, however, until the nineteenth century that mathematicians separated irrationals into algebraic and transcendental parts, and once more undertook scientific study of irrationals. It had remained almost dormant since Euclid. 1872 brought publication of the theories of Karl Weierstrass (by his pupil Kossak), Heine (Crelle, 74), Georg Cantor (Annalen, 5), and Richard Dedekind. In 1869, Méray had taken the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method was completely set forth by Salvatore Pincherle (1880), and Dedekind's has received additional prominence through the author's later work (1888) and 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.

[edit] 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 show 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.

The existence of transcendental numbers^[10] 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.

[edit] Infinity and infinitesimals

The earliest known conception of mathematical infinity appears in the Yajur Veda, an ancient Indian script, 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 c. 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.

Aristotle defined the traditional Western notion of mathematical infinity. He 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, transfinite numbers 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 infinitesimal calculus by Newton and 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 drawing.

[edit] 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 such as Niccolo Fontana Tartaglia and 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. When René Descartes coined the term "imaginary" for these quantities in 1637, he intended it as derogatory. (See imaginary number for a discussion of the "reality" of complex numbers.) A further source of confusion was that the equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \left ( \sqrt{-1}\right )^2 =\sqrt{-1}\sqrt{-1}=-1

seemed capriciously inconsistent with the algebraic identity

Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{a}\sqrt{b}=\sqrt{ab},

which is valid for positive real numbers a and b, and 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{-1}

to guard against this mistake.

The 18th century saw the work of Abraham de Moivre and Leonhard Euler. de Moivre's formula (1730) states:

Failed to parse (Missing texvc executable; please see math/README to configure.): (\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n \theta \,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \cos \theta + i\sin \theta = e ^{i\theta }. \,

The existence of complex numbers was not completely accepted until Caspar Wessel described the geometrical interpretation in 1799. Carl Friedrich Gauss rediscovered and popularized it several years later, 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 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 x² + 1 = 0). His student, Gotthold Eisenstein, studied the type a + bω, where ω is a complex root of x³ − 1 = 0. Other such classes (called cyclotomic fields) of complex numbers derive from the roots of unity x^k − 1 = 0 for higher values of k. This generalization is largely due to Ernst 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 eventually led to the concept of the extended complex plane.

[edit] 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. Goldbach and Riemann's conjectures remain unproven and unrefuted.

Word alternatives[link]

Some numbers traditionally have alternative words to express them, including the following:

• Nothing: 0
• Single: 1
• Pair, couple, brace: 2
• Trio: 3
• Half-Dozen: 6
• Decade: 10
• Dozen: 12
• Baker's dozen: 13
• Score: 20
• Half-century: 50
• Century: 100
• Gross: 144
• Ream: 480 (old measure) 500 (new measure)
• Millenium: 1000
• Great gross: 1728
• "n-figure", as in digit, generally for larger-number ranges, also written without a hyphen; often used in financial discussion. For example

• "five-figure": 10,000 to 99,999 (five digits); ten-thousands
• "six-figure": 100,000 to 999,999 (six digits); hundred-thousands
• "seven-figure": 1,000,000 to 9,999,999 (seven digits); millions

See also[link]

Wikimedia Commons has media related to: Numbers

Notes[link]

References[link]

• Tobias Dantzig, Number, the language of science; a critical survey written for the cultured non-mathematician, New York, The Macmillan company, 1930.
• Erich Friedman, What's special about this number?
• Steven Galovich, Introduction to Mathematical Structures, 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.
• Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, Cambridge University Press, 1910.
• George I. Sanchez, Arithmetic in Maya,Austin-Texas, 1961.

External links[link]

Look up number in Wiktionary, the free dictionary.

Wikiversity has learning materials about Primary mathematics:Numbers

• Nechaev, V.I. (2001), "Number", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=Number 
• Mesopotamian and Germanic numbers
• BBC Radio 4, In Our Time: Negative Numbers
• '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).
• http://planetmath.org/encyclopedia/MayanMath2.html
• "What's the World's Favorite Number?". 2011-06-22. http://www.npr.org/blogs/krulwich/2011/07/22/138493147/what-s-your-favorite-number-world-wide-survey-v1. Retrieved 2011-09-17. ; "Cuddling With 9, Smooching With 8, Winking At 7". 2011-08-11. http://www.npr.org/templates/transcript/transcript.php?storyId=139797360. Retrieved 2011-09-17. 

Maher Zain ماهر زين
Background information
Born	(1981-03-16) March 16, 1981 (age 31) Tripoli, Lebanon
Origin	Sweden
Genres	R&B, Soul music, Pop music, World music, Acoustic music, Nasheed
Occupations	Singer, songwriter, musician, composer, music producer
Instruments	Guitar, piano, keyboard, percussion, synthesizers
Years active	2009–present
Labels	Awakening Records Sony Music
Associated acts	Irfan Makki Mesut Kurtis

Maher Zain (Arabic: ماهر زين‎ ; born March 16, 1981 in Tripoli, Lebanon) is a Muslim Swedish R&B singer, songwriter and music producer of Lebanese origin. His debut album Thank You Allah, an internationally successful album with strong Muslim religious influences, was released in 2009. He released his follow-up album Forgive Me on April 2, 2012.

Career[link]

Beginnings[link]

Maher Zain and his family moved to Sweden when he was eight years old. He completed his schooling there gaining a Bachelors degree in Aeronautical Engineering. After university he entered the music industry in Sweden and linked up with RedOne, the Moroccan-born Swedish producer.^[1]

When RedOne moved to New York in 2006 Maher Zain followed soon after to continue his music industry career in the USA producing for artists such as Kat DeLuna.^[2] On a visit home to Sweden he became engaged once more with his Islamic faith and decided to move away from a career as a music producer to become a singer/songwriter of contemporary R&B music with a strong Muslim religious influence.^[3]

[edit] Breakthrough success and Thank You Allah

In January 2009, Maher Zain began working on an album with Awakening Records. His debut album was Thank You Allah, with 13 songs and two bonus tracks was released on 1 November 2009 with percussion versions and French versions of some tracks released shortly afterwards.^[4]

Zain and Awakening Records successfully used new media such as Facebook, YouTube^[5] and iTunes to promote tracks from the album.^[6][7] In early 2010 his music very quickly gathered a huge online following in Arabic speaking and Islamic countries as well as among young Muslims in western countries.^[8] By the end of 2010 he was the most Googled celebrity in Malaysia for that year.^[7] Malaysia is also the country where he has had the most commercial success. The album Thank You Allah received 8 Platinum awards from Warner Music Malaysia as the highest selling album of 2010.^[9] In 2011, Thank You Allah earned a double platinum award from Sony Music Indonesia.^[1]

Zain sings mainly in English but has released some of his most popular songs in other languages. The song "Insha Allah", for example, is now available in English, French, Arabic, Turkish, Malay and Indonesian versions. Another song, “Allahi Allah Kiya Karo” (“Continuously Saying Allah”), is sung in Urdu and features the Pakistan born Canadian singer Irfan Makki.^[6]

In March 2011, Maher Zain released "Freedom",^[6] a song inspired by the events and the actions of the people taking part in the Arab Spring. In July 2011 he featured on the cover of the UK Muslim lifestyle magazine Emel.^[10]

Zain was featured on Irfan Makki's track "I Believe" from his debut studio album of the same name.^[11] He was also featured on Mesut Kurtis' album, Beloved, in the track named "Never Forget".^{[citation needed]}

Awards[link]

• In January 2010, Maher Zain won Best Religious Song for 'Ya Nabi Salam Alayka', on Nujoom FM, a major Middle East mainstream music station, beating other prominent singers including Hussein Al-Jismi, Mohammed Mounir and Sami Yusuf.^[12]

• Maher Zain was chosen as a Muslim Star of 2011 in a competition organized by Onislam.net.^[13]

Discography[link]

Albums[link]

Singles[link]

• 2010: "Insha Allah"

Videography[link]

• 2009: "Palestine Will Be Free"
• 2010: "Insha Allah"
• 2010: "The Chosen One"
• 2011: "Freedom"
• 2011: "Ya Nabi Salam Alayka"
• 2011: "For the Rest of My Life"
• 2012: "Number One For Me"
• 2012: "So Soon"

Featured in

• 2011: "I Believe" (Irfan Makki feat. Maher Zain) (in Irfan Makki's album I Believe)
• 2011: "Never Forget" (Mesut Kurtis feat. Maher Zain) (in Mesut Kurtis' album Beloved)

References[link]

External links[link]

• Maher Zain Facebook
• Maher Zain on IslamicLyrics.net
• Awakening Music
• Last.fm: Maher Zain page

DJ Wich (also known as Tomáš Pechlák) is a DJ who works the Czech/Slovkia hip-hop scene. He started DJing in 1998.

History[link]

In 1993, DJ Wich met Indy and LA4 on a graffiti scene. In 1998 they formed Indy & Wich,^[1] where Indy performed as a rapper and DJ Wich as producer. They often work with rapper LA4 in live performances.

In 2002, they released their debut album My3. Their second album Hádej kdo… was nominated on Anděl in 2006. The following year they produced The Chronicles of Nomad with rapper Nironic.

In 2008, DJ Wich released his first solo album The Golden Touch with international stars like Lil Wayne, Talib Kweli, M.O.P, Kurupt and more.

He collaborated with Czech artists, including Orion, Hugo Toxxx, W518, LA4 and from the international Rasco, Rytmus, H16, Vec, and Nironic.

Wich streams live shows from (DJ Wich Live TV) with other rappers, live recording, interviews and chat.

DJ Wich made soundtracks for Czech movie Gympl and Ulovit miliardáře which was released on 1 October 2008 with Tomáš Vorel as director.

Awards[link]

• 2001 – Cena Anděl – DJ of the year
• 2002 – Dance Music Awards – My3 – The best album of the year
• 2004 – Music awards of Óčko – Time Is Now – The best Hip Hop/R’N'B album of the year
• 2004, 2005, 2006, 2007, 2008 – Ladder Music Awards – DJ of the year

Discography[link]

• 2001 - Indy & Wich – Cesta Štrýtu
• 2002 - Indy & Wich – My3
• 2003 - Indy & Wich - Ještě pořád
• 2004 - DJ Wich – Time Is Now
• 2004 - DJ Wich – Work Affair Mixtape
• 2006 - Indy & Wich – Hádej Kdo…
• 2007 - DJ Wich presents Nironic: The Chronicles of a Nomad
• 2008 - DJ Wich – The Yearbook Mixtape
• 2008 - DJ Wich – The Golden Touch
• 2009 - Hi-Def & DJ Wich – Human Writes
• 2010 - Dj Wich feat Rasco - The Untouchables: Al Capone's Vault
• 2011 - Dj Wich & Nironic - Nomad 2: The Long Way Home

References[link]

Jay Leno
Leno in July 2008
Birth name	James Douglas Muir Leno
Born	(1950-04-28) April 28, 1950 (age 62)[1] New Rochelle, New York, U.S.[1]
Medium	Television, Film, Stand up
Nationality	American
Years active	1973–present
Genres	Observational comedy, Political satire
Subject(s)	American culture, Everyday life
Influences	Johnny Carson, Robert Klein, Alan King, George Carlin,[2] Don Rickles, Bob Newhart, Rodney Dangerfield
Influenced	Dennis Miller[3]
Spouse	Mavis Leno (1980–present)
Notable works and roles	The Tonight Show with Jay Leno (host, 1992–2009, 2010–) The Jay Leno Show (host, 2009–2010)
Signature
Website	The Tonight Show with Jay Leno
Emmy Awards
Outstanding Variety, Music or Comedy Series 1995 The Tonight Show with Jay Leno

James Douglas Muir "Jay" Leno /ˈlɛnoʊ/ (born April 28, 1950)^[1] is an American stand-up comedian and television host.

From 1992 to 2009, Leno was the host of NBC's The Tonight Show with Jay Leno. Beginning in September 2009, Leno started a primetime talk show, titled The Jay Leno Show, which aired weeknights at 10:00 p.m. (Eastern Time, UTC-5), also on NBC. After The Jay Leno Show was canceled in January 2010 amid a host controversy, Leno returned to host The Tonight Show with Jay Leno on March 1, 2010.^[4]

Early life[link]

James "Jay" Leno was born in New Rochelle, New York, in 1950. His mother, Catherine (née Muir; 1911–1993), a homemaker, was born in Greenock, Scotland, and came to the United States at age 11. Leno's father, Angelo (1910–1994), who worked as an insurance salesman, was born in New York to immigrants from Flumeri, Italy.^[5] Leno grew up in Andover, Massachusetts, and although his high school guidance counselor recommended that he drop out of school, he later obtained a Bachelor's degree in speech therapy from Emerson College, where he started a comedy club in 1973.^[6] Leno's siblings include his late older brother, Patrick, who was a Vietnam veteran^[7] and a lawyer.^[6]

Career[link]

During the 1970s, Leno appeared in minor roles in several television series and films, first in the 1976 episode "J.J. in Trouble" of Good Times and the same year in the pilot of Holmes & Yo-Yo. After an uncredited appearance in the 1977 film Fun with Dick and Jane, he played more prominent parts in 1978 in American Hot Wax and Silver Bears. Other films and television series from that period include Almost Heaven (1978), "Going Nowhere" (1979) from One Day at a Time, Americathon (1979), Polyester (1981), "The Wild One" (1981) from Alice, "Feminine Mistake" (1979) and "Do the Carmine" (1983) from Laverne & Shirley.

[edit] The Tonight Show

Leno in 1993, shortly after becoming host of The Tonight Show

Starting in 1987, Leno was a regular substitute host for Johnny Carson on The Tonight Show. In 1992, he replaced Carson as host^[8] amid controversy with David Letterman, who had been hosting Late Night with David Letterman since 1982 (aired after The Tonight Show), who many had expected to be Carson's successor. The story of this turbulent transition was later turned into a book and a movie. Leno continued to perform as a stand-up comedian throughout his tenure on The Tonight Show.

In 2004, Leno signed a contract extension with NBC which would keep him as host of The Tonight Show until 2009.^[9] Later in 2004, Conan O'Brien signed a contract with NBC under which O'Brien would become the host of The Tonight Show in 2009, replacing Leno at that time.^[10]

During the 2007–08 Writers Guild of America strike, Leno was accused of violating WGA guidelines by writing his own monologue for The Tonight Show. While NBC and Leno claim there were private meetings with the WGA where there was a secret agreement allowing this, the WGA denied such a meeting.^[11] Leno answered questions in front of the Writers Guild of America, West trial committee in February 2009 and June 2009, and when the WGAW published its list of strike-breakers on 11 August 2009, Leno was not on the list.^[12][13]

In 1998, Leno competed in a tag-team match at the WCW's "Road Wild" pay-per-view . In 2001, he voiced The Crimson Chin, a superhero in the Nickelodeon animated series The Fairly OddParents and continues to do so today.

Leno said in 2008 that he was saving all of his income from The Tonight Show and living solely off his income from stand-up comedy.^[14]

On April 23, 2009, Leno checked himself into a hospital with an undisclosed illness.^[15] He was released the following day and returned to work on Monday, April 27. The two subsequently cancelled Tonight Show episodes for April 23 and April 24 were Leno's first in 17 years as host.^[16][17] Initially, the illness that caused the absence was not disclosed, but later Leno told People magazine that the ailment was exhaustion.^[17][18]

Michael Jackson trial[link]

In the 2005 trial of Michael Jackson over allegations of child molestation, Leno appeared as a defense witness (many celebrity defense witnesses had been expected, but Leno was one of the few whose testimony was actually needed). In his testimony regarding a call by the accuser, Leno testified that he never called the police, that no money was asked for, and there was no coaching — but that the calls seemed unusual and scripted.^[19]

Leno in 2006.

As a result, Leno was initially not allowed to continue telling jokes about Jackson or the case, which had been a fixture of The Tonight Show's opening monologue in particular. But he and his show's writers used a legal loophole by having Leno briefly step aside while stand-in comedians took the stage and told jokes about the trial. Stand-ins included Roseanne Barr, Drew Carey, Brad Garrett, and Dennis Miller among others.^[20]

[edit] Succession by Conan O'Brien and The Jay Leno Show

Because Leno's show continued to lead all late-night programming in the Nielsen ratings, the pending expiration of Leno's contract led to speculation about whether he would become a late-night host for another network after his commitment to NBC expired.^[21] Leno left The Tonight Show on Friday May 29, 2009,^[22][23] and Conan O'Brien took over on June 1, 2009.

On December 8, 2008, it was reported that Leno would remain on NBC and move to a new hour-long show at 10 p.m. Eastern Time (9 p.m. Central Time) five nights a week.^[24] This show follows a similar format to The Tonight Show, tapes at the same lot, and retains many of Leno's most popular segments. Late Night host Conan O'Brien was his successor on The Tonight Show.^[25]

Jay Leno's new show, titled The Jay Leno Show, debuted on September 14, 2009. It was announced at the Television Critics Association summer press tour that it would feature one or two celebrities, the occasional musical guest, and keep the popular "Headlines" segments, which would air near the end of the show. First guests included Jerry Seinfeld, Oprah Winfrey (via satellite), and a short sit-down with Kanye West discussing his controversy at the 2009 MTV Video Music Awards.^[26]

[edit] Timeslot conflict and return to The Tonight Show

Wikinews has related news: US TV host Conan O'Brien rejects NBC's offer to switch his show's time slot

In their new roles, neither O'Brien nor Leno succeeded in delivering the viewing audiences the network anticipated. On January 7, 2010, multiple media outlets reported that beginning March 1, 2010, Jay Leno would move from his 10pm weeknight time slot to 11:35pm, due to a combination of pressure from local affiliates whose newscasts were suffering, and both Leno's and O'Brien's poor ratings.^[27][28] Leno's show would be shortened from an hour to 30 minutes. All NBC late night programming would be preempted by the 2010 Winter Olympics between February 15 and February 26. This would move The Tonight Show to 12:05am, a post-midnight timeslot for the first time in its history. O'Brien's contract stipulated that NBC could move the show back to 12:05 a.m. without penalty (a clause put in primarily to accommodate sports preemptions).^[29]

On January 10, NBC confirmed that they would move Jay Leno out of primetime as of February 12 and intended to move him to late night as soon as possible.^[30][31] TMZ reported that O'Brien was given no advance notice of this change, and that NBC offered him two choices: an hour-long 12:05am time slot, or the option to leave the network.^[32] On January 12, O'Brien issued a press release that stated he would not continue with Tonight if it moved to a 12:05am time slot,^[33] saying, "I believe that delaying The Tonight Show into the next day to accommodate another comedy program will seriously damage what I consider to be the greatest franchise in the history of broadcasting. The Tonight Show at 12:05 simply isn’t The Tonight Show."

On January 21, it was announced that NBC had struck a deal with O'Brien. It was decided that O'Brien would leave The Tonight Show. The deal was made that O'Brien would receive a $33 million payout and that his staff of almost 200 would receive $12 million in the departure. O'Brien's final episode aired on Friday, January 22.^[34][35][36] Leno returned as host of The Tonight Show following the 2010 Winter Olympics on March 1, 2010.

On July 1, 2010, Variety reported that total viewership for Jay Leno's Tonight Show had dropped from 5 million to 4 million for the second quarter of 2010, compared to the same period in 2009. Although this represented the lowest second-quarter ratings for the show since 1992, Tonight was still the most-watched late night program, ahead of ABC's Nightline (3.7 million) and David Letterman's Late Show (3.3 million).^[37] Ratings over the following summer, when compared to the same period in 2009 with O'Brien hosting The Tonight Show (including O'Brien's highly rated debut), showed that while total viewership was 12% higher for Leno, viewership in the important "adults aged 18–49" demographic was 23% lower.^[38] NBC ratings specialist Tom Bierbaum commented that due to the host being out of late night television for a period of time and the subsequent 2010 Tonight Show conflict, Leno's ratings fall was "not a surprise at all".^[39]

Public image[link]

Criticism of Leno[link]

Leno on The Tonight Show in 2005

Leno has faced heated criticism and some negative publicity for his perceived role in the 2010 Tonight Show timeslot conflict.^[40][41] Critics have pointed to a 2004 Tonight Show clip, wherein Leno claimed he would allow O'Brien to take over without incident.^[41][42] At the time, Leno stated he didn't want O'Brien to leave for a competing network, adding, "I'll be 59 when [the switch occurs], that's five years from now. There's really only one person who could have done this into his 60s, and that was Johnny Carson; I think it's fair to say I'm no Johnny Carson."^[42] Leno also described The Tonight Show as a dynasty, saying "you hold it and hand it off to the next person. And I don't want to see all the fighting..." At the end of the segment, he said, "Conan, it's yours! See you in five years, buddy!"^[43]

Actor and comedian Patton Oswalt was among the first celebrities to openly voice disappointment with Leno, saying, "Comedians who don't like Jay Leno now, and I'm one of them, we're not like, 'Jay Leno sucks;' it's that we're so hurt and disappointed that one of the best comedians of our generation... willfully has shut the switch off."^[44] Rosie O'Donnell has been among O'Brien's most vocal and vehement supporters,^[45][46] calling Leno a "bully" and his recent actions "classless and kind of career-defining."^[47]

Bill Zehme, the co-author of Leno's autobiography Leading with My Chin, told the Los Angeles Times: "The thing Leno should do is walk, period. He's got everything to lose in terms of public popularity by going back. People will look at him differently. He'll be viewed as the bad guy."^[48]

Support for Leno[link]

NBC Sports head executive and former Saturday Night Live producer Dick Ebersol spoke out against all who had recently mocked Leno, calling them "chicken-hearted and gutless."^[49]

Jeff Gaspin also defended Leno: "This has definitely crossed the line. Jay Leno is the consummate professional and one of the hardest-working people in television. It's a shame that he's being pulled into this."^[48]

Paul Reiser and Jerry Seinfeld are two of the number of celebrities to have voiced support for Leno.^[50][51]

Responding to the mounting criticism, Leno claimed that NBC had assured him that O'Brien was willing to accept the proposed arrangement and then would not let either host out of his contract.^[52] Leno also said that the situation was "all business."^[52] He appeared on the January 28 episode of The Oprah Winfrey Show in an attempt to repair some of the damage done to his public image.^[53][54]

Personal life[link]

Leno with President Barack Obama in March 2009

Leno has been married since 1980 to Mavis Leno; they have no children.^[55]

He is known for his prominent jaw, which has been described as mandibular prognathism.^[56] In the book Leading with My Chin he stated that he is aware of surgery that could reset his mandible, but does not wish to endure a prolonged healing period with his jaws wired shut.

Leno is dyslexic.^[6] He claims to sleep only four to five hours each night.^[57] Leno does not drink or smoke, nor does he gamble.^[58] He spends most of his free time visiting car collections or working in his private garage.^[58]

Leno reportedly earns $32 million each year;^[59] his total net worth is unknown, but has been estimated to be at least $150 million.^[60]

Charity[link]

In 2001, along with his wife, he donated $100,000 to the Feminist Majority's campaign to stop gender apartheid in Afghanistan, to educate the public regarding the plight of women in Afghanistan under Taliban rule. Mavis Leno is on the board of the Feminist Majority.^[61][62]

In 2009, he donated $100,000 to a scholarship fund at Salem State College in honor of Lennie Sogoloff. Mr. Sogoloff gave Leno his start at his jazz club, Lennie's-on-the-Turnpike.^[63]

Love Ride[link]

Since 1985 Jay Leno has been the grand marshal for the Love Ride, a motorcycle charity event which since its founding in 1984 has raised nearly $14 million dollars for charities benefiting muscular dystrophy research and, in 2011, Autism Speaks.^{[citation needed]}

Leno arriving at the 45th Primetime Emmy Awards in his Hispano-Suiza Aero^[64]

Vehicle collection[link]

Leno owns approximately 100 vehicles, not including about 90 motorcycles.^[65] He also has a website called "Jay Leno's Garage," which contains video clips and photos of his automobiles in detail.^[66]

He has a regular column in Popular Mechanics which showcases his car collection and gives advice about various automotive topics, including restoration and unique models, such as his jet-powered motorcycle and solar-powered hybrid. Leno also writes occasional "Motormouth" articles for The Sunday Times,^[67] reviewing high-end sports cars and giving his humorous take on automotive matters.

References[link]