90 (ninety) is the natural number preceded by 89 and followed by 91.

Look up ninety in Wiktionary, the free dictionary.

Interstate 90 is a freeway that runs from Washington to Massachusetts.

In mathematics[link]

• Because 90 is the sum of its unitary divisors (excluding itself), it is a unitary perfect number, and because it is equal to the sum of a subset of its divisors, it is a semiperfect number. 90 is a pronic number. But it is also a nontotient. It is a Perrin number, preceded in the sequence by 39, 51, 68.
• In normal space, the interior angles of a square measure 90 degrees each. Also, in a right triangle, the angle opposing the hypotenuse measures 90 degrees, with the other two angles adding up to 90 for a total of 180 degrees.^[1] Thus, an angle measuring 90 degrees is called a right angle.^[2]
• 90 is divisible by the sum of its base 10 digits, thus it is a Harshad number.

In science[link]

• Ninety is the atomic number of thorium, an actinide. As an atomic weight, 90 identifies an isotope of strontium, a by-product of nuclear reactions including fallout. It contaminates milk.
• The latitude of the North Pole and the South Pole are 90 degrees.

In sports[link]

• Nike Total 90 Apparel is a brand name of football apparel and football equipment from equipment bags to goalkeeper gloves
• Major League baseball bases are 90 feet (27 m) away in distance
• The car number most associated with former NASCAR team owner Junie Donlavey
• The number of minutes in a football (soccer) match.

In Wellness Plans[link]

• P90X is a rigorous work out plan for 90 days with proclaimed incredible results.

See also[link]

Mathematics portal

References[link]

1. ^ http://www.sparknotes.com/testprep/books/newsat/chapter20section4.rhtml
2. ^ Friedman, Erich. What's Special About This Number?

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. 

Giorgio Moroder
Giorgio Moroder
Background information
Birth name	Hansjörg Moroder
Also known as	Giorgio
Born	(1940-04-26) 26 April 1940 (age 72) Urtijëi, Italy
Origin	Urtijëi, Italy
Genres	Dance, disco, electronic, electronic dance, Hi-NRG, Eurodisco, Italo disco, pop, rock
Occupations	Singer, songwriter, record producer
Instruments	Synthesizers
Years active	1965–present
Labels	Oasis
Associated acts	Donna Summer, Freddie Mercury, Limahl
Website	www.moroder.net

Hansjörg "Giorgio" Moroder,^[1][2] named on record sleeves often only Giorgio (born 26 April 1940, Urtijëi, South Tyrol, Italy) is an Italian (of Ladin ethnicity^[3]) record producer, songwriter and performer based in Los Angeles. When in Munich in the 1970s, he started his own record label called Oasis Records, which several years later became a subdivision of Casablanca Records. He collaborated with Donna Summer during the era of disco (including "Love to Love You Baby" and "I Feel Love") and is the founder of the former Musicland Studios in Munich, which was used as a recording studio by artists including the Electric Light Orchestra, Led Zeppelin, Queen and Elton John.

In addition to producing several hits with Donna Summer, Moroder also produced a number of electronic disco hits for The Three Degrees, two albums for Sparks, a handful of songs on Bonnie Tyler's album Bitterblue as well as her 1985 single "Here She Comes" and a score of songs for performers including David Bowie, Irene Cara, Madleen Kane, Melissa Manchester, Blondie, Japan, and France Joli.

Music career[link]

Moroder made his first steps in music in Berlin by releasing a few singles under the name "Giorgio" beginning in 1966, singing in Italian, Spanish, English, and German. He came to prominence in 1969, when his recording "Looky Looky", released on Ariola Records, was awarded a gold disc in October 1970.^[4] He then began making a name for himself in studios around Germany in the early 1970s. Often collaborating with lyricist Pete Bellotte, Moroder had a number of hits in his own name including "Son of My Father" in 1972 before releasing the synthesizer-driven From Here to Eternity, a notable chartbuster in 1977, and in the following year releasing "Chase", the theme from the film Midnight Express. These songs achieved some chart success in the United Kingdom, the United States, and across Europe, and everywhere disco-mania was spreading. The full film score for Midnight Express won him his first Academy Award for best film score in 1978. In 1979, Moroder released his album E=MC². Text on the album's cover stated that it was the "first electronic live-to-digital album." He also released three albums between 1977–1979 under the name Munich Machine.

In 1984, Moroder worked with Philip Oakey of The Human League to make the album Philip Oakey & Giorgio Moroder; which was a UK singles chart hit with "Together in Electric Dreams", title track to the 1984 film Electric Dreams. 1985 saw him collaborating with Kajagoogoo frontman Limahl for their worldwide hit "Neverending Story". In 1986, Moroder collaborated with his protégé Harold Faltermeyer (of "Axel F.") and lyricist Tom Whitlock to create the score for the film Top Gun (1986) and Berlin's "Take My Breath Away". "Chase" was also used as an entrance theme for wrestling's group The Midnight Express. In 1987, Moroder produced Falco's song "Body Next to Body".

In 1997, Moroder and Donna Summer won the Grammy Award for Best Dance Recording for the song "Carry On".

On 20 September 2004 Moroder was honored at the Dance Music Hall of Fame ceremony, held in New York, when he was inducted for his many outstanding achievements and contributions as producer. In 2005, he was given the title of Commendatore by the then President of the Italian Republic, Carlo Azeglio Ciampi. On September 5, 2010 Moroder received the Great Order of Merit of the South Tyrol.

Film work[link]

Throughout much of his musical career, Moroder has been involved in soundtrack work for various motion pictures.

In 1980, he composed and produced two film soundtrack albums: the first for Foxes and the second for American Gigolo. A double album, of the Foxes soundtrack was released on the disco label Casablanca Records which includes Donna Summer's hit single "On the Radio", which Moroder both produced and co-wrote. The American Gigolo soundtrack featured the Moroder-produced Blondie's "Call Me", a US and UK number one hit. All singles from the album also went to number two for five weeks on the disco/dance charts.^[5] In 1982 he wrote the soundtrack of the movie Cat People, including the hit single "Putting Out Fire" featuring David Bowie.

In 1983, Moroder produced the soundtrack for the film Scarface. During its initial release, the album was only available in a few countries and strictly through import in the United States. Notable Moroder-produced tracks included "Scarface (Push It to the Limit)" by Paul Engemann, "Rush Rush" by Debbie Harry, and "She's on Fire" by Amy Holland. In 2006, the soundtrack was featured in the game Scarface: The World Is Yours, which is based on the film and also includes some previously-unreleased instrumentals by Moroder.

In 1984, Moroder compiled a new restoration and edit of the silent film Metropolis (1927) and provided it with a contemporary soundtrack. This soundtrack includes seven pop music tracks from Pat Benatar, Jon Anderson, Adam Ant, Billy Squier, Loverboy, Bonnie Tyler and Freddie Mercury. He also integrated the old-fashioned intertitles into the film as subtitles as a means of improving continuity, and he also increased the film's frame to 24 frames a second. Since the original speed was unknown this choice was controversial. Known as the "Moroder version", it sparked debate among film buffs, with outspoken critics and supporters of the film falling into equal camps.^[6]

Moroder also scored other films in the 1970s and 1980s, including Midnight Express, Flashdance, The Never Ending Story, Thief of Hearts, Electric Dreams, Cat People and Over the Top.

In 2002, he wrote the score for Leni Riefenstahl's final film, Impressionen unter Wasser, a marine documentary.^[7]

In an interview in May 2012, Moroder said he is collaboration on a little "rap" on the upcoming album of Daft Punk.^[8][9]

Video games[link]

His score for "Scarface" has been used in the video game Grand Theft Auto III while Moroder's "From Here to Eternity" and "Chase" were used in the Sony PlayStation Portable and PlayStation 2 game Grand Theft Auto: Liberty City Stories. "Together In Electric Dreams", a collaborative effort of Giorgio Moroder and Philip Oakey (of The Human League) features in Grand Theft Auto: Vice City Stories.

Sporting events and other media[link]

Moroder wrote the official theme songs, "Reach Out", for the 1984 Los Angeles Olympics, and "Hand in Hand", for the 1988 Seoul Olympics and "Un'estate italiana" for the 1990 Football World Cup. "The Chase" is now also used as the theme bumper-music for the US AM talk radio program Coast to Coast AM. The BBC used Moroder's film soundtrack "Ivory Tower" as theme song during the broadcast of the motor-racing event Formula 1 Grand Prix, during the late 1980s and the early/mid 1990s. He also composed the song "Forever Friends" for the 2008 Summer Olympics in Beijing.

Other[link]

In the late 1980s, Moroder collaborated with Claudio Zampolli to create the Cizeta supercar.

Awards[link]

Moroder has won three Academy Awards: Best Original Score for Midnight Express (1978); Best Song for "Flashdance...What a Feeling", from the film Flashdance (1983); and Best Song for "Take My Breath Away", from Top Gun (1986).

Moroder also won two of his three Grammy Awards for "Flashdance": Best Album Of Original Score Written For A Motion Picture Or A Television Special and Best Instrumental Composition, for the track "Love Theme from Flashdance".

Notable collaborations[link]

{{columns-list|4|

• María Conchita Alonso
• Adam Ant
• Pat Benatar
• Edoardo Bennato
• Berlin
• Big Trouble
• Blondie
• Daft Punk
• David Bowie* [[* The Three Degrees
• Bonnie Tyler

}}

Discography[link]

The discography of Giorgio Moroder includes thirteen studio albums and ten soundtracks. This list is not representative of Moroder's work, as he is most successful as music producer.

Studio albums[link]

Singles[link]

• 1965 Baby Courreges / Warum hoer' ich nichts von dir (as Rock Romance)
• 1966 Stop / Believe In Me
• 1966 Glaub An Mich / Stop
• 1966 Bla, Bla Diddley / How Much Longer Must I Wait, Wait (as Giorgio and The Morodians)
• 1967 Lilly Belle / Love's Morning Land
• 1968 Moody Trudy / Stop
• 1968 Yummy, Yummy, Yummy / Make Me Your Baby (as Giorgio)(as George)
• 1968 Cinnamon / Reesy-Beesy
• 1968 Monja / Raggi Di Sole (as George)
• 1969 Luky, Luky [Looky, Looky] / Senza Te, Senza Me [Happy Birthday] (as George)
• 1969 Looky, Looky / Happy Birthday
• 1969 Mah Nà Mah Nà / Doo-Bee-Doo-Bee-Doo
• 1970 Lina Con La Luna (Moody Trudy) / Risi E Bisi (Reesy Beesy) (as George)
• 1970 Mony Mony (Muny Muny Muny) / Tempo D'amore (Make Me Your Baby)(as George)
• 1970 Arizona Man / Sally Don't You Cry
• 1970 Arizona Man (Jerk) / So Young (Slow)
• 1970 America, America / Rhythm Of Love (as Spinach)
• 1971 Action Man (part 1) / Action Man (part 2) (as Spinach)
• 1971 Underdog / Watch Your Step
• 1971 I'm Free Now / Son Of My Father
• 1971 London Traffic / Everybody Join Hands
• 1972 (Sweet Sixteen) You Know What I Mean / Knockin' On Your Door
• 1972 Tu Sei Mio Padre [Son Of My Father] / Non Ci Sto [Underdog]
• 1972 Son Of My Father (part 1) / Son Of My Father (part 2)
• 1972 Lord (Release Me) / Tears (as Children Of The Mission)
• 1972 Today's A Tomorrow (You Worried 'Bout Yesterday) / Pauline
• 1972 Take It, Shake It, Break My Heart / Spanish Disaster
• 1972 The Future Is Past / Blue Jean Girl
• 1973 Lonely Lovers' Symphony / Crippled Words
• 1973 Heaven Helps The Man (Who Helps Himself) / Sandy
• 1973 Hilf Dir Selbst/Geh Zu Ihm
• 1974 Marrakesh / Nostalgie
• 1974 Lie, Lie, Lie / Collico
• 1974 Born To Die (Mono) / Born To Die (Stereo)(as Giorgio)
• 1974 Born To Die / Strongest Of The Strong (as Giorgio's Common Cause)
• 1975 Bricks and Mortar / It's A Shame
• 1975 Rock Me To My Soul / Dark and Deep and In between (as Giorgio's Common Cause)
• 1975 Einzelgänger / Liebes Arie (as Einzelgänger)
• 1976 Einzelgänger / Good Old Germany (as Einzelgänger)
• 1976 Take Five / Enterprise (as MLS (Musicland Set))
• 1976 Knights In White Satin
• 1976 I Wanna Funk With You Tonite / Oh, L'Amour
• 1977 Let The Music Play / Oh, L'Amour
• 1977 From Here To Eternity / Too Hot To Handle
• 1977 From Here To Eternity / Utopia - Me Giorgio
• 1977 Get On The Funk Train (part I) / Get On The Funk Train (part II) (as Munich Machine)
• 1978 A Whiter Shade of Pale / It's All Wrong (But It's Alright) (as Munich Machine)
• 1978 Love's In You (Love's In Me) / I Can't Wait (as Giorgio & Chris)
• 1979 Party Light / Bolectro (as Munich Machine)
• 1979 Baby Blue / If You Weren't Afraid
• 1979 If You Weren't Afraid / E=MC2
• 1980 Night Drive / The Apartment
• 1983 Scarface (Push It To The Limit) / Tony's Theme (as Giorgio Moroder & Paul Engemann)
• 1984 Together In Electric Dreams / Together In Electric Dreams Instrumental (as Giorgio Moroder & Philip Oakey)
• 1984 The Duel / Madeline's Theme
• 1984 Reach Out / Reach Out (Instrumental)(as Giorgio Moroder & Paul Engemann)
• 1984 American Dream / Too Hot To Touch (Instrumental) (as Giorgio Moroder & Paul Engemann)
• 1985 Shannon's Eyes / Shannon's Eyes (Instrumental) (as Giorgio Moroder & Paul Engemann)
• 1985 Night Time Is The Right Time / Baby Blue [1985 Remix]
• 2012 Giorgio's New Dancing Shoes (from the CD: The 84th Academy Awards)

Soundtracks[link]

Sampling and other uses[link]

• Bronski Beat have covered "I Feel Love" and "Love to Love You Baby" for their debut album The Age of Consent (1984).
• Madonna has sampled "I Feel Love" ( a production by Moroder, sung by Donna Summer ) in the early 90's.
• "On Fire", the second single from rapper Lil Wayne's seventh studio album Rebirth; contains allusions from Amy Holland's song "She's on Fire" and was inspired in its entirety by Scarface.^[11]
• "Push It", the second single from rapper Rick Ross' debut album Port of Miami; samples "Scarface (Push It to the Limit)" and the story of the video has a very similar theme to the film Scarface. It was produced by J. R. Rotem.
• His song "Tears" was sampled and used as the basis of the DJ Shadow song "Organ Donor" on his 1996 album Endtroducing...... Canadian Hip Hop group Swollen Members sampled the song in "Fuel Injected" and "Meltdown". It also appears on the song "Tragedy" by RZA. The main melody and chord progression form the basis of "Marz" by folk musician John Grant and "Only Light" by Australian ska band The Cat Empire.
• The Hip Hop-duo Mobb Deep used a sample from the song "Tony's Theme" in their song G.O.D. Pt. III.
• His Song "E=MC2" was sampled and used for J. Dilla's song of the same title.
• One of his early compositions, "Doo-Bee-Doo-Bee-Doo" from 1969, would be featured for many years in silent sketches on The Benny Hill Show, as part of a medley that also included "Mah Nà Mah Nà", a 4/4 adaptation of Ludwig van Beethoven's "Für Elise", and "Gimme Dat Ding".
• The theme from Midnight Express was sampled by hip-hop duo OutKast for their song "Return of the Gangsta", and by hip-hop producer J Dilla for "Phantom of the Synths", a beat later used by MF DOOM for "Gazzillion Ear" and by Jay Electronica for "Dimethyltryptamine".
• Moroder's opening theme from the film Scarface is sampled by Nas and Mobb Deep for the track "It's Mine".
• "Leopard Tree Dream" from Cat People is sampled by Cannibal Ox in the song "Iron Galaxy."
• "The Legend Of Babel" theme From the "metropolis" soundtrack was covered By DJ Dado.
• British electronica musician Little Boots covered "Love Kills", which was written in collaboration with Freddie Mercury.
• "Future Lovers", a song from American recording artist Madonna's 2005 album Confessions on a Dance Floor, has a bass line inspired by Donna Summer's Moroder-produced hit "I Feel Love". Furthermore, Madonna opened her 2006 Confessions Tour with a medley of "Future Lovers" and "I Feel Love".
• For the "Chase" instrumental, see that article
• Suns of Arqa's album "Technomor" includes the track "Moroder Vibe" which contains elements of "I Feel Love".

See also[link]

• List of number-one dance hits (United States)
• List of artists who reached number one on the U.S. Dance chart
• Cizeta Moroder V16 supercar project

References[link]

This biographical article needs additional citations for verification. Please help by adding reliable sources. Contentious material about living persons that is unsourced or poorly sourced must be removed immediately, especially if potentially libelous or harmful. (November 2010)

External links[link]

• discog.info: Giorgio Moroder discography
• Giorgio Moroder discography at Discogs
• Giorgio Moroder at the Internet Movie Database