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. 

Keri Hilson
Keri Hilson performing at Supafest in Australia, April 2011
Background information
Birth name	Keri Lynn Hilson
Born	(1982-12-05) December 5, 1982 (age 29) Decatur, Georgia, U.S.
Genres	R&B, pop
Occupations	Singer, songwriter, dancer, actress
Years active	2001–present
Labels	Zone 4, Mosley Music, Interscope
Associated acts	The Clutch, Polow da Don, Timbaland, Chris Brown, Akon, Kanye West
Website	kerihilson.com

Keri Lynn Hilson (born December 5, 1982) is an American singer, songwriter and actress. She was born and raised in Decatur, Georgia, and spent most of her youth working with producer Anthony Dent, as a songwriter and background vocalist for several R&B and hip hop artists. By the age of 14, Hilson had secured a record deal with the girl group D'Signe, who later disbanded. She attended Emory University in Atlanta, while she continued writing songs for artists, including Britney Spears and Mary J. Blige, with the production and songwriting team, The Clutch. In 2006, Hilson signed with American rapper and producer Timbaland's record label, Mosley Music. Her breakthrough came the following year after appearing on Timbaland's single, "The Way I Are", which topped charts around the world.

Hilson's debut studio album, In a Perfect World..., was released in 2009, and debuted at number four on the US Billboard 200 chart, eventually being certified gold by the Recording Industry Association of America (RIAA). The album spawned the hit singles, "Knock You Down", "Turnin Me On" and "I Like". It also earned Hilson two Grammy Award nominations for Best New Artist and Best Rap/Sung Collaboration for "Knock You Down". Her second studio album, No Boys Allowed, was released in 2010 and was considerably less successful than its predecessor; however, it included the hit single, "Pretty Girl Rock". Hilson's achievements include a BET Award, MOBO Award, NAACP Image Award and two Soul Train Music Awards. Apart from her work in music, she has pursued a career in acting. Hilson made her acting debut in the romantic comedy film, Think Like a Man, in 2012.

Biography[link]

1982–2007: Early life and career beginnings[link]

Keri Lynn Hilson was born on December 5, 1982 in Decatur, Georgia.^[1][2] She was raised in an African-American neighborhood and came from a middle class family.^[3] Hilson and her siblings did not attend a school within the neighborhood because her parents cared a lot for their education, stating that "they shipped us off for an hour to get to school every day. It was just that important for us to maintain the life that they had created".^[3] From the age of 12, Hilson wanted to pursue a musical career, after watching television talent shows, Star Search and Showtime at the Apollo.^[1] Her mother then hired a piano teacher to give her piano lessons; however, Hilson wanted to become a singer and therefore "converted those sessions into vocal lessons, accompanying the teacher on piano".^[1] By the age of 14, Hilson had secured a record deal with the girl group D'Signe, who later disbanded.^[4][1]

Hilson spent most of her teenage years working with producer Anthony Dent, as a songwriter and background vocalist for the likes of Usher, Ludacris, Kelly Rowland, Ciara and Polow da Don.^[1] After graduating from high school, Hilson attended Emory University in Atlanta for three years and studied a course in theater.^[3] She continued working as a songwriter and background vocalist during her schooling years.^[1] Hilson eventually stopped working with Dent, and began working more with Polow da Don.^[1] She also became a member of the songwriting and production team, The Clutch.^[1] Together, they wrote Mary J. Blige's "Take Me as I Am" (2006), Omarion's "Ice Box" (2006), Ciara's "Like a Boy" (2006), The Pussycat Dolls' "Wait a Minute" (2007) and Britney Spears' "Gimme More" (2007).^[1][4] In 2006, Polow introduced Hilson to American rapper and producer Timbaland, who then signed her to his record label, Mosley Music Group.^[1] In November 2006, Hilson was featured on American rapper Lloyd Banks' single, "Help", for his album Rotten Apple (2006).^[5]

The following year, Hilson was featured on three singles; in June, on Timbaland's song "The Way I Are", the second single from his album Shock Value (2007); on Rich Boy's "Good Things" with Polow da Don; and thereafter, in December, on Timbaland's song "Scream" with Nicole Scherzinger, the fifth single from Shock Value.^[6] Out of the three singles, "The Way I Are" achieved the most success, topping nine charts around the world.^{[7][8][9][10][11]}

[edit] 2008–09: In a Perfect World...

Hilson in April 2009

In 2008, Hilson appeared in the music videos for Usher's single, "Love in This Club", and Ne-Yo's single, "Miss Independent".^[12][13] Her debut studio album, In a Perfect World..., was released on March 24, 2009.^[14] The album contained pop-oriented R&B songs.^[15] It was initially planned to be released the previous year; however, the album was delayed several times because Hilson wanted to make sure it came out to her liking, stating "I'm just a perfectionist. I have had many [release dates], but it's all for a good cause. The label wants to make sure the album has its proper release, and I'm thankful for that. Very grateful, even though fans look at it like it's a bad thing."^[16] In the US, In a Perfect World... debuted at number four on the Billboard 200 chart, and number one on the Top R&B/Hip-Hop Albums chart, with first-week sales of 94,000 copies.^[17][18] By October 2009, the album was certified gold by the Recording Industry Association of America (RIAA).^[19] Its first single, "Energy", released in May 2008, peaked within the top eighty on the US Billboard Hot 100 chart.^[20] It reached the top fifty in the United Kingdom, and peaked the highest in New Zealand at number seven, and was certified gold in the country.^[10][21][22] Hilson also featured on three singles in 2008; in June, on rapper Nas' song "Hero"; in September, on Kardinal Offishall's "Numba 1 (Tide Is High)"; and thereafter, in October, on Chris Brown's song "Superhuman".^[23][6][24]

"Return the Favor", featuring Timbaland, served as the second international single from Hilson's In a Perfect World... album, while "Turnin Me On" featuring Lil Wayne, was released as the second US single. The latter achieved commercial success, reaching number 15 on the Billboard Hot 100, and number two on the Hot R&B/Hip-Hop Songs chart, eventually being certified platinum in the US.^[20][19] The next single, "Knock You Down" featuring Kanye West and Ne-Yo, achieved more success, reaching number three on the Hot 100 and number one on the Hot R&B/Hip-Hop Songs chart, eventually being certified two times platinum.^[20][19] The song also reached the top ten in Canada, Ireland, The Netherlands, New Zealand and the United Kingdom.^{[20][25][21][10]} "Make Love", "Slow Dance", and "Change Me" featuring Akon, were released as the album's fourth, fifth and sixth US singles, respectively.

Hilson, along with Gym Class Heroes, Gorilla Zoe and T-Pain, were supporting acts on Lil Wayne's I Am Music Tour in North America.^[26] At the 2009 BET Awards, she was nominated in four categories, including Best New Artist, Best Female R&B Artist, as well as Viewer's Choice and Best Collaboration for "Turnin Me On" with Lil Wayne.^[27] She eventually won the Best New Artist category.^[28] In 2009, Hilson was featured on five singles; "Number One" with R. Kelly, "She Don't Wanna" with Asher Roth, "Everything, Everyday, Everywhere" with Fabolous, "Medicine" with Plies, and on the remix of Sean Paul's song, "Hold My Hand".^[29][6] Hilson was nominated for Breakthrough Artist and Favorite Soul/R&B Female Artist at the 2009 American Music Awards, but lost to Beyoncé Knowles.^[30] The re-released edition of In a Perfect World..., in January 2010, included the single "I Like", which reached number one in Germany, Poland and Slovakia, and peaked within the top ten in Austria, Norway and Switzerland.^{[31][32][33][34]} At the 52nd Grammy Awards, Hilson was nominated for Best New Artist and Best Rap/Sung Collaboration for "Knock You Down" with Kanye West and Ne-Yo.^[35]

[edit] 2010–present: No Boys Allowed

Hilson performing at the 2010 VH1 Divas Salute the Troops concert

During the first half of 2010, Hilson guest featured on rapper Trina's single, "Million Dollar Girl", and on T.I.'s single, "Got Your Back".^[6] In April 2010, it was made known that she had replaced Jennifer Hudson as the new face of the cosmetics and personal care products company, Avon.^[36] "Breaking Point" was released as the first single from Hilson's second studio album, No Boys Allowed, in September 2010; it only appeared on the US Hot R&B/Hip-Hop Songs Chart at number 44.^[20] On December 2, 2010, Hilson was among one of the many female artists who performed at the VH1 Divas Salute the Troops concert, where she sang "Turnin Me On", "Knock You Down", "Pretty Girl Rock", and a duet with American country music duo, Sugarland, on the Aretha Franklin song, "Think".^[37] No Boys Allowed was released on December 21, 2010.^[38] Described as a "girl power album", No Boys Allowed was primarily of the R&B and pop genres.^[39][40] Despite the album's title, Hilson stated "it's not about excluding men. It's more about women understanding that there comes a time in your life when you want a man. A real man. A grown up. Not a boy. And that's not a bad thing."^[41] Speaking of the songs on the album, she explained "I write from a female perspective, but I'm also telling men what women are really thinking and feeling about them".^[41] In the US, No Boys Allowed debuted at number 11 on the Billboard 200, and number seven on the Top R&B/Hip-Hop Albums chart, with first-week sales of 102,000 copies.^[42][43] Although it sold 8,000 copies more than her debut album, In a Perfect World..., it failed to match that album's debut chart position of number four.^[42] As of February 2011, No Boys Allowed has sold 205,500 copies in the United States.^[44]

Hilson achieved success with the album's second single, "Pretty Girl Rock", which reached number 24 on the Billboard Hot 100 and number four on the Hot R&B/Hip-Hop Songs Chart, eventually being certified platinum.^[20][19] The single reached the top twenty in Germany and New Zealand, and top thirty in Austria and Slovakia.^{[45][21][46][47]} "One Night Stand" featuring Chris Brown, and "Lose Control (Let Me Down)" featuring Nelly, were released as the album's third and fourth singles, respectively. In 2011, she was featured on British rapper Chipmunk's single "In the Air", for his album Transition (2011).^[48] In April 2011, Hilson, along with many other R&B and hip hop acts, traveled to Australia to be part of its biggest urban music festival, Supafest.^[49] At the 2011 BET Awards, Hilson was nominated for Best Female R&B Artist and Video of the Year for "Pretty Girl Rock".^[50] In July 2011, she was a supporting act on the second leg of Lil Wayne's I Am Still Music Tour in North America, before embarking on her first headlining European tour in October.^[51][52]

2012–present: Acting debut and third studio album[link]

During an interview with 106 & Park on October 25, 2011, Hilson revealed that she has already begun recording her third studio album.^[53] She explained, "There's some people in the world that aren't gon' like this album coming from me. As I was experiencing the world, there were things that I was also experiencing on a personal front, and in my music it's coming out. There's a specific few people who ain't gon' like to listen to this album. Very emotional, I'll say that."^[53] She further explained in an interview with The Boombox that she's working on a new sound for the album, and described it as a mixture of her first two albums with some surprising elements thrown in.^[54] Hilson also said a release date has not been confirmed yet, saying "I just write about my experiences and keep the release dates far, far from me. When I feel that I have [good material] is when I will give it to the label. But I'm not quite there yet. I'm still working."^[54] Hilson made her acting debut in the romantic comedy film, Think Like a Man, released on April 20, 2012.^[55][56] She will also star alongside Vin Diesel in the sci-fi action film, Riddick (2013), but it is unknown what role Hilson will play.^[57]

Artistry[link]

Musical style and influences[link]

Hilson's music is generally R&B and pop. Her debut album, In a Perfect World..., consists of a pop/R&B style, with elements of electro music.^[15][58] Many of the album's themes deal with relationships, physical attraction and lovemaking.^[59] Hilson said, "lyrically it's a very vulnerable album. You know, I definitely didn't want to paint myself as perfect on this project. Instead, I wanted it to be something that was very relatable, especially to women".^[60] Her second album, No Boys Allowed, displays a wide variety of styles, including pop, R&B, hip hop, soul, acoustic, electronic and reggae.^{[40][61][62][63]} Hilson describes the album as "more self assured. It's a lot more aggressive".^[64] Kristin Macfarlane of The Daily Post noted that the album is "pure girl power and about being sexually confident, and confident as a woman; not putting up with crap relationships and getting your way".^[65] Hilson sings with "smooth vocals".^[66] Following the release of her debut album, Sophie Bruce of BBC Music expressed that, "there's no denying Hilson has a great voice, but it lacks the feisty edge of [Nicole] Scherzinger, [Mary J.] Blige or [Beyoncé] Knowles".^[67] Mark Nero of About.com commented, "Keri's voice, while strong, isn't particularly distinctive or memorable".^[68] Hilson has stated that her musical influences come from her father's side of the family.^[60] In addition, she cites Michael Jackson and Lauryn Hill as inspirations.^[69][70]

Philanthropy[link]

Besides her career in music, Hilson has also been involved with several charities. In January 2010, she teamed up with Akon on the charity single, "Oh Africa".^[71] All proceeds of the single benefited underprivileged African youth.^[71] The following month, Hilson joined the extended list of artists during the recording session of the single, "We Are the World 25 for Haiti", to benefit victims of the 2010 Haiti earthquake.^[72] On May 25, 2010, she performed at the Virginia Stand Up! A Call to Action benefit concert, organized by Chris Brown to help with continued relief efforts in Haiti.^[73] On October 21, 2010, Hilson attended Collins Academy High School in Chicago as part of the Get Schooled National Challenge and Tour, a "program aimed at increasing high school and college graduation rates and promoting the importance of education".^[74][75] A year later, she received a Get Schooled Award for her involvement with the program.^[76] In 2011, Hilson contributed in the fight against AIDS, by posing in retail-clothing company H&M's celebrity-driven collection for Fashion Against AIDS.^[77] 25% of all sales from the collection were donated to the Designers Against Aids charity.^[77] Hilson contributed to the It Gets Better Project, a project which aims to prevent suicide among LGBT youth.^[78][79]

In June 2011, Hilson became an ambassador for MTV Staying Alive, a foundation which raises awareness and prevention of HIV and AIDS.^[80] In September 2011, she lent her support to US First Lady Michelle Obama's Let's Move! campaign, which aims to combat childhood obesity.^[81] She made t-shirts for the campaign that read "Pretty Girls Sweat", and exercised with the young women at The Educational Alliance's Sirovich Senior Center in New York on September 24.^[81] During Hilson's visit in Africa in November 2011, she stopped by an orphanage to visit her young fans who had lost their families during the 2010–2011 Ivorian crisis.^[82] She gave each orphan a "care" package, which consisted of bookbags, t-shirts, sunglasses, notebooks and an electric keyboard.^[82] In April 2012, Hilson teamed up with DoSomething.org to encourage young people to join in the nationwide Epic Book Drive by bringing books to local people in need.^[83] In a press release, she stated: "Joining DoSomething.org's newest effort to provide for people in need with Epic Book Drive was the easiest decision ever!. I truly believe providing access to those who want to learn is the key to bettering lives around the country. And helping kids find healthy outlets, such as books, has always been a passion of mine. So my involvement here is a natural fit."^[83]

Discography[link]

• In a Perfect World... (2009)
• No Boys Allowed (2010)

Filmography[link]

Tours[link]

Headlining

• European Tour (2011)

Supporting

• I Am Music Tour (2009) (with Lil Wayne)
• I Am Still Music Tour (2011) (with Lil Wayne)

Awards and nominations[link]

References[link]

External links[link]

	Book: Keri Hilson
Wikipedia books are collections of articles that can be downloaded or ordered in print.

Media related to Keri Hilson at Wikimedia Commons

• Official website
• Keri Hilson at the Internet Movie Database

Phil Collins
Phil Collins performing in Perugia, Italy, 1996
Background information
Birth name	Philip David Charles Collins
Born	(1951-01-30) 30 January 1951 (age 61) Chiswick, Middlesex, England, United Kingdom
Genres	Progressive rock, rock, pop, pop rock, soft rock, jazz fusion, R&B, big band
Occupations	Singer-songwriter, musician, actor
Instruments	Vocals, drums, percussion, guitar, piano, keyboards
Years active	1968–2011
Labels	Virgin, Atlantic
Associated acts	Genesis, Brand X, Flaming Youth, Philip Bailey, Eric Clapton, the Phil Collins Big Band, Frida Lyngstad, Brian Eno
Website	philcollins.co.uk

Philip David Charles "Phil" Collins, LVO (born 30 January 1951)^[1] is an English singer-songwriter, drummer, pianist and actor best known as a drummer and vocalist for British progressive rock group Genesis and as a solo artist.

Collins sang the lead vocals on several chart hits in the United Kingdom and the United States between 1975 and 2010, either as a solo artist or with Genesis. His singles, sometimes dealing with lost love, ranged from the drum-heavy "In the Air Tonight", dance pop of "Sussudio", piano-driven "Against All Odds", to the political statements of "Another Day in Paradise".

Collins's professional music career began as a drummer, originally in a band called The Real Thing with Andrea Bertorelli, who later became his first wife. Collins played drums and shared lead vocals (with Brian Chatton) in Flaming Youth which recorded one album, (Ark II). In 1970, he took over drums for Genesis, which had already recorded two albums. In Genesis, Collins originally supplied backing vocals for front man Peter Gabriel, singing lead on only two songs: "For Absent Friends" from 1971's Nursery Cryme album and "More Fool Me" from Selling England by the Pound, which was released in 1973. Following Gabriel's departure in 1975, Collins became the group's lead singer.

His solo career, heavily influenced by his personal life, brought both himself and Genesis commercial success. According to Atlantic Records, Collins's total worldwide sales as a solo artist, as of 2000, were 150 million.^[2] Collins has won numerous music awards throughout his career, including seven Grammy Awards, five Brit Awards—winning Best British Male three times, an Academy Award, and two Golden Globes for his solo work.^[3][4] He was inducted into the Rock and Roll Hall of Fame as a member of Genesis in 2010.

Collins is one of only three recording artists (along with Paul McCartney and Michael Jackson) who have sold over 100 million albums worldwide both as solo artists and (separately) as principal members of a band.^[5] When his work with Genesis, his work with other artists, as well as his solo career is totalled, Collins had more top 40 hits on the Billboard Hot 100 chart during the 1980s than any other artist.^[6] In 2008, Collins was ranked the 22nd most successful artist on the "The Billboard Hot 100 Top All-Time Artists".^[7]

Early life[link]

Collins was born in Chiswick, London, the son of Winifred M. "June" (née Strange), a theatrical agent, and Greville Philip Austin Collins, an insurance agent.^[8][9] He was given a toy drum kit for Christmas when he was five. Later, his uncle made him a makeshift one that he used regularly. As Collins grew older these were followed by more complete sets bought by his parents.^[10] He practiced by playing alongside the television and radio, and never learned to read and write conventional musical notation; instead, he uses a system he devised himself.^[11]

Early career[link]

His professional training began at 14 when he entered Barbara Speake Stage School.^[12] He began a career as a child actor and model, and won his first major role as the Artful Dodger in the London production of Oliver!. He was an extra in the Beatles' A Hard Day's Night, one of hundreds of screaming teenagers during the TV concert sequence and seen fleetingly in a close-up. He was also in Chitty Chitty Bang Bang as one of the children who storms the castle at the end of the film, but it was cut.^[13] He also auditioned for the role of Romeo in Romeo and Juliet (1968),^[14] a role won by fellow Artful Dodger actor Leonard Whiting.^[15] Collins was among the last three finalists for the role of I.Q. on the American children's television show The Bugaloos (he lost out to English actor/musician John McIndoe).^[16]

Despite the beginnings of an acting career, Collins continued to gravitate towards music. While attending Chiswick Community School he formed a band called the Real Thing and later joined the Freehold. With the latter group, he wrote his first song titled "Lying Crying Dying".^[17]

Collins's first record deal came as drummer for Hickory, who changed their name to Flaming Youth by the time of their sole album, Ark 2 (1969).^[18] A concept album inspired by the recent media attention surrounding the moon landing, Ark 2 (with Ronnie Caryl, Brian Chatton and Gordon (Flash) Smith), failed to make much commercial success despite positive critical reviews. Melody Maker featured the album as "Pop Album of the Month", describing it as "adult music beautifully played with nice tight harmonies".^[19] The album's main single, "From Now On", failed on the radio. After a year of touring, band tensions and the lack of commercial success dissolved the group. In 1970, the 19-year old Collins played percussion on the George Harrison song "The Art of Dying". Harrison credited him in the liner notes to the remastered CD version of the album released in 2000.^[20]

Genesis era[link]

In 1970, Collins answered a Melody Maker classified ad for "...a drummer sensitive to acoustic music, and acoustic twelve-string guitarist".^[21] Genesis placed the ad after having already lost three drummers over two albums.^[22] The audition occurred at the home of Peter Gabriel's parents. Prospective candidates performed tracks from the group's second album, Trespass (1970). Collins arrived early, listened to the other auditions while swimming in Gabriel's parents' pool, and memorised the pieces before his turn.^[23]

Collins won the audition. Nursery Cryme was released a year later. Although his role remained primarily that of drummer and backing vocalist for the next five years, he twice sang lead vocals: once on "For Absent Friends" (from Nursery Cryme) and once on "More Fool Me" (from Selling England by the Pound).

In 1974, while Genesis were recording the concept album The Lamb Lies Down on Broadway, Brian Eno (who is credited with "Enossification" for electronic vocal effects on the track "Grand Parade of Lifeless Packaging") needed a drummer for his album Taking Tiger Mountain (By Strategy).^[24] Collins was sent to fill the gap, and played drums in lieu of payment for Eno's work with the band. Collins later contributed drums to the Brian Eno 1975, and 1977 art rock releases Another Green World and Before and After Science.

In 1975, following the final tour supporting the album The Lamb Lies Down on Broadway, Gabriel left the group to pursue a solo career. Collins became lead vocalist after a lengthy but ultimately fruitless search for Gabriel's replacement (where he sang back-up with the over 400 hopefuls that reportedly auditioned).^[25] In the short term, the group recruited former Yes and King Crimson drummer Bill Bruford to play drums during live shows, although Collins continued to play during longer instrumental sections. Bruford's drumming can be heard on the track "The Cinema Show" on the live album Seconds Out. He was soon replaced by ex-Frank Zappa band member Chester Thompson, who became a mainstay of the band's live line-up. Collins, however, continued to play drums on all of the group's studio recordings.

The first album with Collins as lead vocalist, 1976's A Trick of the Tail, reached the American Top 40, and peaked high as No.3 on the UK charts. Said Rolling Stone, "Genesis has managed to turn the possible catastrophe of Gabriel's departure into their first broad-based American success."^[26] Following the recording of Genesis's next album Wind and Wuthering guitarist Steve Hackett left the group to pursue his own solo career. The group decided to continue as a trio for recording with Mike Rutherford playing guitar and bass in the studio, although the lineup was regularly augmented by Chester Thompson and American guitarist Daryl Stuermer for concert tours.

Collins simultaneously performed in a jazz fusion group called Brand X. The band recorded their first album, Unorthodox Behaviour, with Collins as drummer, but because Genesis was Collins's priority, there were several Brand X tours and albums without him. Collins credits Brand X as his first use of a drum machine as well as his first use of a home 8-track tape machine.^[27]

Collins also performed on Steve Hackett's first solo album, Voyage of the Acolyte, on which he sang lead vocals and played drums. As the decade closed, Genesis began a shift from their progressive rock roots and toward more accessible, radio-friendly pop-rock music. The album ...And Then There Were Three... featured their first UK Top 10 and US Top 40 single, "Follow You Follow Me".^[28][29]

	"Dance on a Volcano" (1976) noicon The first track from Genesis's A Trick of the Tail was Collins's début as the group's full-time lead singer. A progressive rock track, it contrasts with the style of his later work.
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In the 1980s, while Collins developed as a songwriter and established a parallel career as a solo artist, Genesis recorded a series of highly successful albums including Duke, Abacab, Genesis, and Invisible Touch. The latter album's title track reached No.1 on the American Billboard singles chart, the only Genesis song to do so. The group received an MTV "Video of the Year" nomination in 1987 for the single "Land of Confusion" (which featured puppet caricatures created by the British satirical team Spitting Image) but lost out to Peter Gabriel's solo hit, "Sledgehammer".^[30] Reviews were generally positive, with Rolling Stone's J. D. Considine stating, "every tune is carefully pruned so that each flourish delivers not an instrumental epiphany but a solid hook."^[31]

Collins left Genesis in 1996 to focus on his solo career.^[17] The last studio album with him as the lead singer was 1991's We Can't Dance. He and Gabriel reunited with other Genesis members in 1999 to re-record "The Carpet Crawlers" for Genesis's Turn It on Again: The Hits. When in the mid-2000s discussions of a possible Genesis reunion arose, Collins stated that he would prefer to return as the drummer, with Gabriel handling the vocals.^[32] Eventually Turn It On Again: The Tour was announced for 2007, with the Collins/Rutherford/Banks line-up.

In March 2010, Phish guitarist Trey Anastasio was asked to pay tribute to Genesis, one of his favourite bands, upon being inducted into the Rock & Roll Hall of Fame. In addition to Anastasio's speech, Phish appeared and performed two Genesis songs, "Watcher of the Skies" and "No Reply At All". Collins and his Genesis band-mates (minus Peter Gabriel) attended the ceremony but they did not perform.

Solo career[link]

Early solo recordings (1981–1983)[link]

The dominant theme running through Collins's early solo recordings (although never specifically mentioned in his songs) was the acrimonious breakdown of his first marriage and then-recent divorce. Two songs he wrote on the Genesis album Duke, "Please Don't Ask", and the Top 20 hit "Misunderstanding", dealt with failed relationships. One year earlier, he had played drums and sung backing vocals on John Martyn's Grace and Danger, an album whose main theme is also marriage break-up. With the recording of his first solo album, Face Value, Collins attributed his divorce as his main influence,^[33] as can be inferred from songs such as "If Leaving Me Is Easy".

	"In the Air Tonight" noicon "In the Air Tonight" from Face Value (1981), was the first single of Collins' solo career.
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Collins made his live debut as a solo performer, appearing at the invitation of producer Martin Lewis at the Amnesty International benefit show, The Policeman's Other Ball at the Theatre Royal in London in September 1981, performing two songs from Face Value including "In the Air Tonight" and "The Roof is Leaking" accompanying himself on piano. Face Value became a surprise international success topping the charts in at least seven countries and hitting the top ten of the Billboard 200 eventually going triple-platinum in the US. Hits from the album included "In the Air Tonight", "I Missed Again" and "If Leaving Me Is Easy". In 1982, he produced ABBA member Frida's solo album, Something's Going On, which helped to spawn the title track, "I Know There's Something Going On", which became a hit.

Much like Face Value, many of the songs from Collins's 1982 follow-up album, Hello, I Must Be Going!, came from Collins's marital problems with his first wife such as "I Don't Care Anymore" and "Do You Know and Do You Care". Collins’s early albums had a dark presence, usually heavy on the drums. Regarding Face Value, he says, "I had a wife, two children, two dogs, and the next day I didn't have anything. So a lot of these songs were written because I was going through these emotional changes."^[34] There were occasional poppier influences–Face Value's "Behind the Lines", for example, was a jazzy remake of a Genesis song he co-wrote. Face Value was a critical and multi-platinum success, and saw Collins’s profile increase further. Hello, I Must Be Going! gave him a UK No.1 for his cover of the Supremes' "You Can't Hurry Love". The album went triple-platinum in the United States, like its predecessor. The Supremes' cover was his first Top 10 US hit (it also hit the Top 10 of Billboard's Adult Contemporary chart). The album also reached No.2 on the UK album chart, spending well over a year there.^[28]

Two years before, Collins had played drums on Peter Gabriel's third self-titled record (often referred to as Melt), the first record to feature the "gated reverb" sound, which was used on the song "Intruder". Gabriel reportedly "didn't want any metal on the record" and asked Collins to leave his cymbals at home, to concentrate on the sound of his kit more heavily than usual. Studio engineer Hugh Padgham augmented the drum sound by using a microphone normally intended for studio communication rather than recording and feeding it through a signal processor called a noise gate. This allowed the reverberation added to the drums to be suddenly cut off before it naturally decayed. The result was the arresting "gated reverb" which became Collins signature sound. This was the same 'big drum sound' used on such songs as "In The Air Tonight", "Mama" by Genesis, and Frida's "There's Something Going On".

Mid-career (1984–1992)[link]

	"Against All Odds" noicon "Against All Odds" from the soundtrack of the same name (1984), is a piano-driven ballad. "Another Day in Paradise" noicon A sample of "Another Day in Paradise" from ...But Seriously (1989). The ballad was written to bring attention to the problem of homelessness
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Collins changed his musical style with the release of the ballad, "Against All Odds (Take a Look at Me Now)", which was the theme song to the movie of the same name in 1984. The more pop-friendly and radio-accessible single became Collins's first solo single to reach number-one on the Billboard Hot 100.^[29] Later that year, Collins contributed to production on Earth, Wind & Fire singer Phillip Bailey's debut album, Chinese Wall, collaborating with Bailey on the hit duet, "Easy Lover". Collins featured on Band Aid's "Do They Know It's Christmas", he played drums and sang on the song. Collins released his most successful album, No Jacket Required, in early 1985. It contained the hits "Sussudio", "One More Night", "Don't Lose My Number", and "Take Me Home", as well as the less known yet equally robust "Who Said I Would", and "Only You Know and I Know". The album featured Sting, Helen Terry and ex-bandmate Peter Gabriel as backing vocalists. He also recorded the successful song "Separate Lives", a duet with Marilyn Martin, and a US number one, for the movie White Nights.^[29] Collins had three US number one songs in 1985, the most by any artist that year.^[29] No Jacket Required went on to win several Grammy awards including Album of the Year.

No Jacket Required received criticism that the album was too safe, despite its upbeat reviews and commercial success. A positive review by David Fricke of Rolling Stone ended, "After years on the art-rock fringe, Collins has established himself firmly in the middle of the road. Perhaps he should consider testing himself and his new fans' expectations next time around."^[35] "Sussudio" also drew criticism for sounding too similar to Prince's "1999", a charge that Collins did not deny.^[36] Nevertheless, the album went straight to #1 in the US and UK. In 1985, Collins was invited by Bob Geldof to perform at the Live Aid charity event. Collins had the distinction of being the only performer to appear at both the UK concert at Wembley Stadium and the US concert at JFK Stadium in Philadelphia. He accomplished this by performing early in the day at Wembley as both a solo artist and alongside Sting, then transferring to a Concorde flight to the US enabling him to perform his solo material, and drum for Led Zeppelin and Eric Clapton in Philadelphia. While being a guest on major artists' hit recordings, Collins continued to enjoy solo success even while on tour with Genesis, besides from his number-one duet with Marilyn Martin in 1986, Collins would score two more hits from movies with the singles, "Two Hearts" (No.1 US, No.6 UK), and "Groovy Kind of Love" (No.1 UK, No.1 US), the latter two from the soundtrack of his feature film, Buster.^[28][29]

In 1989, Collins produced another successful album, ...But Seriously, featuring the anti-homelessness anthem "Another Day in Paradise", with David Crosby on backing vocals. (Collins later went on to co-write, sing and play on the song "Hero" on Crosby's 1993 album Thousand Roads.) "Another Day in Paradise" went to Number 1 on the Billboard Charts at the end of 1989 and won Collins Best British Single at the 1990 Brit Awards, and the 1991 Grammy Award for Record of the Year.^[37][38] In the process, it became the last No.1 US pop hit of the 1980s. The album ...But Seriously became the first No.1 US album of the 1990s and the best-selling album of 1990 in the UK. Other songs included "Something Happened on the Way to Heaven" (No.4 US, No.15 UK), "Do You Remember?" (not released in the UK, but a No.4 hit in the US), and "I Wish It Would Rain Down" (the latter featuring Eric Clapton on guitar) (No.3 US, No.7 UK).^[28][29] Songs about apartheid and homelessness demonstrated Collins’s turn to politically-driven material. This theme recurred on his later albums. A live album, Serious Hits... Live!, followed. In September 1990, Collins performed "Sussudio" at the 1990 MTV Video Music Awards in Los Angeles.^[39]

Later solo work and Genesis reunion (1993–2008)[link]

Collins's record sales began to drop with the 1993 release of Both Sides, a largely experimental album that, according to Collins, included songs that "were becoming so personal, so private, I didn't want anyone else's input".^[40] Featuring a less polished sound and fewer up-tempo songs than his previous albums, Both Sides was a significant departure. Collins used no backing musicians, performed all the vocal and instrumental parts at his home studio, and used rough vocal takes for the final product. The album was not well received by radio. Its two biggest hits were "Both Sides of the Story" and "Everyday". Collins worked on the album completely independently of his record company, and took them by surprise when he delivered them a completed album that they were unaware he was making.

Collins officially parted ways with Genesis in 1996 to focus on his solo career (Genesis would produce one album without Collins—...Calling All Stations...—before going on hiatus). Collins attempted a return to pop music with Dance into the Light, which Entertainment Weekly reviewed by saying that "(e)ven Phil Collins must know that we all grew weary of Phil Collins".^[41] It included minor hits such as the title track and the Beatles-inspired "It's in Your Eyes". Although the album went Gold in the US, it sold considerably less than his previous albums. Despite this, the subsequent tour regularly sold out arenas.

In 1996, Collins formed the Phil Collins Big Band. With Collins as drummer, the band performed jazz renditions of various Collins and Genesis hits. The Phil Collins Big Band did a world tour in 1998 that included a performance at the Montreux Jazz Festival. In 1999, the group released the CD A Hot Night in Paris including big band versions of "Invisible Touch", "Sussudio", and the more obscure "The Los Endos Suite" from A Trick of the Tail.

Phil Collins's star on the Hollywood Walk of Fame, 6834 Hollywood Blvd

A compilation album ...Hits was released in 1998 and sold very well, returning Collins to multi-platinum status in America. The album's sole new track, a cover of the Cyndi Lauper hit "True Colors", received considerable play on US Adult Contemporary stations while peaking at No.2.^[42] Some of Collins's earlier hits (e.g. "I Missed Again", "If Leaving Me Is Easy", etc.) and other successes were not included in this compilation.

Collins's next single, "You'll Be in My Heart", from the Disney animated movie Tarzan, spent 19 weeks at No.1 on Billboard's Adult Contemporary chart – the longest time ever up to that point. The song won Collins an Academy Award for Best Song. It was his third nomination in the songwriters' category, after being nominated in 1985 and 1989. Collins was awarded a star on the Hollywood Walk of Fame, on 16 June 1999.^[43]

Collins performing in Barcelona, Spain in July 2004

In 2002 Collins released Testify. Metacritic's roundup of album reviews found this record to be the worst-reviewed album at the time of its release, though it has since been "surpassed" by three more recent releases.^[44] The album's "Can't Stop Loving You" (a Leo Sayer cover) was yet another #1 Adult Contemporary smash hit for Collins. Testify sold 140,000 copies in the United States by year's end, although a successful worldwide tour followed.^[45]

That same year Collins accepted an invitation to drum for the "house band" at a concert celebrating Queen Elizabeth II's Golden Jubilee. In 2003 announced his last solo tour^[46] – the "First Final Farewell Tour", a tongue-in-cheek reference to the multiple farewell tours of other popular artists. In 2006 he worked with Disney on a Broadway production of Tarzan, a musical which received generally mixed reviews. In 2007 Collins reunited with his Genesis bandmates Tony Banks and Mike Rutherford for Turn It On Again: The Tour, a tour of Europe and North America. During the tour Genesis performed at the Live Earth concert at Wembley Stadium. Following the band's performance, presenter Jonathan Ross had to apologise to viewers watching the televised version as Collins had used a swear word while singing "Invisible Touch".

[edit] Going Back and retirement (2009–2011)