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. 

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

History[link]

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

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

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

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

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

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

Awards[link]

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

Discography[link]

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

References[link]

Hugh Jackman
Jackman at the Sydney premiere for Real Steel, September 2011
Born	Hugh Michael Jackman (1968-10-12) 12 October 1968 (age 43) Sydney, New South Wales, Australia
Occupation	Actor
Years active	1994–present
Spouse	Deborra-Lee Furness (1996–present)

Hugh Michael Jackman (born 12 October 1968) is an Australian actor and producer who is involved in film, musical theatre, and television.

Jackman has won international recognition for his roles in major films, notably as action/superhero, period and romance characters. He is known for his role as Wolverine in the X-Men film series, as well as for his leads in Kate & Leopold, Van Helsing, The Prestige, Australia, and Real Steel. Jackman is a singer, dancer, and actor in stage musicals, and won a Tony Award for his role in The Boy from Oz.

In November 2008, Open Salon named Jackman one of the sexiest men alive.^[1] Later that same month, People magazine named Jackman "Sexiest Man Alive."^[2]

A three-time host of the Tony Awards, winning an Emmy Award for one of these appearances, Jackman also hosted the 81st Academy Awards on 22 February 2009.^[3]

Early life[link]

Jackman was born in Sydney, New South Wales, the youngest of five children of English parents Chris Jackman and Grace Watson, and the second child to be born in Australia (he also has a younger half-sister, from his mother's re-marriage).^[4] One of his paternal great-grandfathers was Greek.^[5] His parents divorced when he was eight, and he remained with his accountant father and siblings, while his mother moved back to England.^[6] As a child, Jackman liked the outdoors, spending a lot of time at the beach and on camping trips and vacations all over Australia. He wanted to see the world: "I used to spend nights looking at atlases. I decided I wanted to be a chef on a plane. Because I'd been on a plane and there was food on board, I presumed there was a chef. I thought that would be an ideal job."^[7]

Jackman went to primary school at Pymble Public School and later attended the all-boys Knox Grammar School, where he starred in its production of My Fair Lady in 1985, and became the captain of the school in 1986.^[8] Following graduation, he spent a gap year working at Uppingham School in England.^[9] On his return, he studied at the University of Technology, Sydney, graduating in 1991 with a BA in Communications.^[10] In his final year of university, he took a drama course to make up additional credits. The class did Václav Havel's The Memorandum with Jackman as the lead.^[4] He later commented, "In that week I felt more at home with those people than I did in the entire three years [at university]".^[11]

After obtaining his BA, Jackman completed the one-year course "The Journey" at the Actors' Centre in Sydney.^[4] About studying acting full-time, he stated, "It wasn't until I was 22 that I ever thought about my hobby being something I could make a living out of. As a boy, I'd always had an interest in theater. But the idea at my school was that drama and music were to round out the man. It wasn't what one did for a living. I got over that. I found the courage to stand up and say, 'I want to do it'."^[7] After completing "The Journey", he was offered a role on the popular soap opera Neighbours but turned it down^[12] to attend the Western Australian Academy of Performing Arts of Edith Cowan University in Perth, Western Australia, from which he graduated in 1994.^[13]

Jackman has said he "always loved acting but when I started at drama school I was like the dunce of the class. It just wasn’t coming right to me. Everyone was cooler, everyone seemed more likely to succeed, everyone seemed more natural at it and in retrospect I think that is good. I think it is good to come from behind as an actor. I think it is good to go into an audition thinking 'Man I’ve got to be at my best to get this gig.'"^[14]

Career[link]

Early work[link]

On the night of his final Academy graduation performance, Jackman received a phone call offering him a role on Correlli: "I was technically unemployed for thirteen seconds." Correlli, devised by Australian actress Denise Roberts, was a 10-part drama series on ABC, Jackman's first major professional job, and where he met his future wife Deborra-Lee Furness: "Meeting my wife was the greatest thing to come out of it."^[7] The show lasted only one season.

After Correlli Jackman went on the stage in Melbourne. In 1996, Jackman played Gaston in the local Walt Disney production of Beauty and the Beast, and Joe Gillis in Sunset Boulevard.^[4] During his stage musical career in Melbourne, he starred in the 1998 Midsumma festival cabaret production Summa Cabaret. He also hosted Melbourne's Carols by Candlelight and Sydney's Carols in the Domain.

Jackman's early film work includes Erskineville Kings and Paperback Hero (1999), and his television work includes Law of the Land, Halifax f.p., Blue Heelers, and Banjo Paterson's The Man from Snowy River.

[edit] Oklahoma!

Jackman became known outside of Australia in 1998, when he played the leading role of Curly in the Royal National Theatre's acclaimed stage production of Oklahoma!, in London's West End.^[4] The performance earned him an Olivier Award nomination for Best Actor in a Musical. Jackman said "I totally felt like it can't get any better than this. On some level that production will be one of the highlights of my career."^[7] He also starred in the 1999 film version of the same stage musical, which has been screened in many countries.

[edit] X-Men

Jackman at the X-Men Origins: Wolverine premiere, April 2009.

In 1999, Jackman was cast as Wolverine in Bryan Singer's X-Men (2000), replacing Dougray Scott. His co-stars included Patrick Stewart, James Marsden, Famke Janssen, and Ian McKellen. According to a CBS interview in November 2006, Jackman's wife Deborra-Lee Furness told him not to take the role, a comment she later told him she was glad he ignored.

Wolverine was tough for Jackman to portray because he had few lines, but a lot of emotion to convey in them. To prepare, he watched Clint Eastwood in the Dirty Harry movies and Mel Gibson in Road Warrior. "Here were guys who had relatively little dialogue, like Wolverine had, but you knew and felt everything. I'm not normally one to copy, but I wanted to see how these guys achieved it."^[7] Jackman was adamant about doing his own stunts for the movie. "We worked a lot on the movement style of Wolverine, and I studied some martial arts. I watched a lot of Mike Tyson fights, especially his early fights. There's something about his style, the animal rage, that seemed right for Wolverine. I kept saying to the writers, 'Don't give me long, choreographed fights for the sake of it. Don't make the fights pretty."^[7]

Jackman also had to get used to wearing Wolverine's claws. "Every day in my living room, I'd just walk around with those claws, to get used to them. I've got scars on one leg, punctures straight through the cheek, on my forehead. I'm a bit clumsy. I'm lucky I didn't tell them that when I auditioned."^[7]

Jackman, at 6 feet 2.5 inches (1.89 m),^[15] stands a foot taller than Wolverine, who is said in the original comic book to be 5 feet 3 inches (1.60 m).^[16] Hence, the filmmakers were frequently forced to shoot Jackman at unusual angles or only from the waist up to make him appear shorter than he actually is, and his co-stars wore platform soles. Jackman was also required to add a great deal of muscle for the role, and in preparing for the fourth film in the series, he bench-pressed over 300 pounds.^[17] An instant star upon the film's release, Jackman later reprised his role in 2003's X2: X-Men United, 2006's X-Men: The Last Stand, and 2009's X-Men Origins: Wolverine. He also cameos as Wolverine in 2011's X-Men: First Class.

2001[link]

Jackman starred as Leopold in the 2001 romantic comedy film Kate & Leopold, a role for which he received a Best Actor Golden Globe nomination.^[4] Jackman plays a Victorian English duke who accidentally time-travels to 21st-century Manhattan, where he meets Kate (Meg Ryan), a cynical advertising executive. In 2001, Jackman also starred in the action/drama Swordfish with John Travolta and Halle Berry. This was the second time Jackman worked with Berry, and the two have worked together twice more in the X-Men movies.

He hosted an episode of Saturday Night Live in 2001.^[18]

2002–2011 (Stage)[link]

Local advertising for the musical The Boy from Oz starring Jackman in New York City (2004)

In 2002, Jackman sang the role of Billy Bigelow in the musical Carousel in a special concert performance at Carnegie Hall with the Orchestra of St. Luke's.

In 2004, Jackman won the Tony Award and the Drama Desk Award for Outstanding Actor in a Musical for his 2003–2004 Broadway portrayal of Australian songwriter and performer Peter Allen in the hit musical The Boy from Oz, which he also performed in Australia in 2006.^[4] In addition, Jackman hosted the Tony Awards in 2003, 2004, and 2005, garnering positive reviews. His hosting of the 2004 Tony Awards earned him an Emmy Award for Outstanding Individual Performer in a Variety, Musical or Comedy program.

Jackman co-starred with Daniel Craig on Broadway at the Schoenfeld Theatre in a limited engagement of the play A Steady Rain, which ran from 10 September 2009 to 6 December 2009.^[19]

He returned to Broadway in a new show, Hugh Jackman, Back on Broadway at the Broadhurst Theatre, which began performances on 25 October 2011 and concluded on 1 January 2012.^[20]

2003–2008 (Films)[link]

After 2003's X2: X-Men United, Jackman played the title role of monster killer Gabriel Van Helsing in the 2004 film Van Helsing.^[4] Jackman and the film were noted in Bruce A. McClelland's book "Slayers and Their Vampires: A Cultural History of Killing the Dead".

Jackman was one of the choices to play James Bond in 2006's Casino Royale, but eventually lost out to Daniel Craig.^[21] Jackman starred in the 2006 film The Prestige, directed by Christopher Nolan and co-starring Christian Bale, Michael Caine, and Scarlett Johansson. As Robert Angier, Jackman portrayed a magician who built up a rivalry with contemporary Alfred Borden in attempt to one-up each other in the art of deception. Jackman stated that his main reason for doing The Prestige was to work with the musician David Bowie, who played scientist Nikola Tesla.

Hugh Jackman and Ryan Reynolds (right) at the X-Men Origins: Wolverine premiere in Tempe, Arizona (2009)

Jackman portrayed three different characters in Darren Aronofsky's science-fiction film The Fountain: Tommy Creo, a neuroscientist, who's torn between his wife, Izzi (Rachel Weisz) who is dying of a brain tumor, and his work at trying to cure her; Captain Tomas Creo, a Spanish Conquistador in 1532 Seville; and a future astronaut, Tom, travelling to a golden nebula in an eco-spacecraft seeking to be reunited with Izzi. Jackman said The Fountain was his most difficult film thus far due to the physical and emotional demands of the part.

Jackman also starred in Woody Allen's 2006 film Scoop opposite Scarlett Johansson. That year he also reprised the role of Wolverine in X-Men: The Last Stand. He rounded out 2006 with two animated films: Happy Feet, directed by George Miller, in which he voiced the part of Memphis, an emperor penguin; and Flushed Away, where Jackman supplied the voice of a rat named Roddy who ends up being flushed down a family's toilet into the London sewer system. Flushed Away co-starred Kate Winslet and Ian McKellen (Jackman's fourth time working with him).

In 2007, Jackman produced and guest-starred in the television musical-dramedy series Viva Laughlin, which was canceled by CBS after two episodes. Jackman's 2008 movies included Deception (which he starred in and produced), Uncle Jonny, and Australia.

[edit] Australia

In 2008, director Baz Luhrmann cast Jackman to replace Russell Crowe as the male lead in his much-publicized epic film, Australia, which co-starred Nicole Kidman. The movie was released in late November 2008 in Australia and the U.S.

Jackman played a tough, independent cattle drover, who reluctantly helps an English noblewoman in her quest to save both her philandering husband's Australian cattle station and the half-caste Aboriginal child she finds there.

Of the movie, Jackman said, "This is pretty much one of those roles that had me pinching myself all the way through the shoot. I got to shoot a big-budget, shamelessly old-fashioned romantic epic set against one of the most turbulent times in my native country's history, while, at the same time, celebrating that country's natural beauty, its people, its cultures.... I'll die a happy man knowing I've got this film on my CV."^[22]

2009 to present[link]

Jackman's X-Men sequel film, X-Men Origins: Wolverine, opened in 2009.

Jackman had a one-man show at the Curran Theatre in San Francisco from 3–15 May 2011.^[23] The production was a mixture of his favorite Broadway and Hollywood musical numbers, backed by a 17-piece orchestra, from shows including Oklahoma and The Boy from Oz. The show had a run-time of approximately 100 minutes, and also included slide shows of Jackman's youth, family, and work, as well as some one-on-one interaction with the audience. Jackman was backed by fellow musical theatre veterans Merle Dandridge and Angel Reda.^[24][25]

Future projects[link]

• Director Lee Daniels has confirmed Jackman has joined the cast of his upcoming film Selma, a film about Martin Luther King Jr. and Lyndon Baines Johnson and the civil rights marches.^[26]
• On 27 January 2011, DreamWorks announced that Jackman will star in the film Rise of the Guardians, scheduled to be released on 21 November 2012.^[27]
• Jackman has been cast in Movie 43, an upcoming 2012 ensemble comedy.^[28]
• Jackman will star as Jean Valjean in the film Les Misérables, an adaptation of the musical.^[29][30][31]
• Jackman will reprise his role as Logan/Wolverine from the X-Men film series in the 2013 film "The Wolverine"^[32]
• Jackman will feature in the film "The Greatest Showman on Earth"^[33]

Production company[link]

In 2005, Jackman joined with longtime assistant John Palermo to form a production company, Seed Productions, whose first project was Viva Laughlin in 2007. Jackman's wife Deborra-Lee Furness is also involved in the company, and Palermo had three rings made with a "unity" inscription for himself, Furness, and Jackman.^[34] Jackman said, "I'm very lucky in the partners I work with in my life, Deb and John Palermo. It really works. We all have different strengths. I love it. It's very exciting."^[35]

The Fox-based Seed label has grown in size to include execs Amanda Schweitzer, Kathryn Tamblyn, Allan Mandelbaum and Joe Marino, with Alana Free operating the Sydney-based production office whose goal is to mount modest-budget films to harness local talent in Jackman's home country.

Other interests[link]

Charity work[link]

Jackman aboard the amphibious assault ship USS Kearsarge in New York (2006)

As a philanthropist, Jackman is a longtime proponent of microcredit – the extension of very small loans to prospective entrepreneurs in impoverished countries. He is a vocal supporter of Muhammad Yunus, microcredit pioneer and the 2006 Nobel Peace Prize winner.^[36][37][38]

Jackman is a global advisor of the Global Poverty Project, for which he narrated a documentary;^[39] and he and the project's founder Hugh Evans visited the UN for the cause in 2009.^[40] Jackman hosted a preview of the Global Poverty Project Presentation in New York together with Donna Karran, Lisa Fox and his wife Deborra-Lee.^[41] He is also a World Vision ambassador and participated in the climate week NYC ceremony on 21 September 2009.^[42][43]

Jackman supports The Art of Elysium^[44] and the MPTV Fund Foundation,^[45] and he and his wife Deborra-Lee Furness are patrons of the Bone Marrow Institute in Australia.^[46] Jackman also narrated the 2008 documentary about global warming, The Burning Season.^[47]

Jackman also uses his Twitter account for charity. On 14 April 2009 Jackman posted on his Twitter page that he would donate $100,000 to one individual's favorite non profit organization.^[48] On 21 April 2009 he revealed his decision to donate $50,000 to Charity:Water and $50,000 to Operation of Hope.^[49][50]

Hugh Jackman and Daniel Craig made a unique place for themselves in the history of Broadway Cares/Equity Fights AIDS fundraising 8 December 2009, when it was announced that they had raised $1,549,953 in the 21st annual Gypsy of the Year competition, from six weeks of curtain appeals at their hit Broadway drama, A Steady Rain.^[51]

Sports and other activities[link]

Jackman and Furness at Mumbai International Airport to attend FICCI – Frames film convention, 2011^[52]

Jackman has shown keen interest in sports. In high school, he played rugby union and cricket, took part in high jumping and was on the swimming team.^[4] He enjoys basketball and kayaking.^[53] He has expressed an interest in football, committing his support to Norwich City FC.^[54] In the United States, Jackman supports the Philadelphia Union of Major League Soccer, attending a game at PPL Park in June 2010.^[55] On 22 June 2011, Jackman again attended a home Union match against Sporting KC, sitting in front of the Sons of Ben supporters section, nicknamed "The River End".

Jackman is a longtime fan and supporter of the Manly-Warringah Sea Eagles, a NRL club based in Sydney's north.^[56] He sang the national anthem at the 1999 NRL Grand Final.^[57]

Jackman also guest starred on the 19 September 2011 edition of WWE Monday Night Raw, assisting Zack Ryder in a win over WWE United States Champion Dolph Ziggler. Jackman helped "Long Island Iced Z" get the 3 count in a non-title match by punching the champion in the jaw whilst the referee was not looking.^[58][59]

Jackman plays the piano,^[60] does yoga,^[61] and has been a member of the School of Practical Philosophy since 1992.^[62]

Personal life[link]

Jackman married Deborra-Lee Furness on 11 April 1996. They met on Correlli, an Australian television series. Jackman personally designed an engagement ring for Furness, and their wedding rings bore the Sanskrit inscription "Om paramar mainamar", translated as "we dedicate our union to a greater source".^[63]

Furness had two miscarriages,^[64] following which she and Jackman adopted two children, Oscar Maximillian (born 15 May 2000)^[65] and Ava Eliot (born 10 July 2005).^[66]

Filmography[link]

Awards and nominations[link]

Awards[link]

Jackman's signature at Grauman's Chinese Theatre

• 1997 Variety Club Award for Best Actor in a Musical – Sunset Boulevard
• 1998 Mo Award for Best Actor in a Musical – Sunset Boulevard
• 1999 Australian Movie Convention, Australian Star of the Year
• 2000 Film Critics Circle of Australia Awards for Best Actor Erskineville Kings
• 2000 Saturn Award for Best Actor – X-Men
• 2001 Academy of Science Fiction, Fantasy & Horror Films - X-Men
• 2004 Drama Desk Award for Outstanding Actor in a Musical – The Boy from Oz
• 2004 Theatre World Award – The Boy from Oz
• 2004 Broadway Audience Award – The Boy from Oz
• 2004 Drama League Award for Distinguished Performance of the Year – The Boy from Oz
• 2004 Outer Critics Circle Award for Best Actor in a Musical – The Boy from Oz
• 2004 TDF-Astaire Award for Best Male Dancer in Theatre – The Boy from Oz
• 2004 Theater Fan's Choice Award for Best Leading Actor in a Musical – The Boy from Oz
• 2004 Tony Award for Best Leading Actor in a Musical – The Boy from Oz
• 2004 New York International Independent Film & Video Festival – Short Film Award for Best Actor – "Making the Grade"
• 2004 Australian Showbusiness Ambassador of the Year
• 2005 Emmy Award for Outstanding Individual Performance in a Variety or Music Program – 58th Annual Tony Awards Ceremonies
• 2006 ShoWest Award for Male Star of the Year
  

  

  Jackman and Halle Berry visiting with sailors and Marines during the opening day of Fleet Week New York 2006
• 2006 Mo Award for Australian Performer of the Year
• 2008 WAAPA – Chancellor's Alumni Award for Excellence, UTS Towering Achievement Award
• 2008 Australian Dance Award for Outstanding Performance in a Stage Musical – The Boy from Oz
• 2008 Australian Film Institute Award Readers' Choice
• 2008 People Magazine's Sexiest Man Alive Award
• 2008 Australian GQ Man of the Year
• 2008 Teen Choice Award for Choice Actor in an Action Adventure X-Men Origins: Wolverine
• 2009 Jackman had his hand and footprint ceremony on the Hollywood Walk of Fame on 21 April 2009.^[68]
• 2009 Best Performance By A Human Male at the 2009 Spike Video Game Awards as Wolverine in X-Men Origins: Wolverine
• 2010 People's Choice Award for Favorite Action Star – X-Men Origins: Wolverine
• 2011 Scream Award for Best Cameo - X-Men: First Class
• 2012 People's Choice Award for Favorite Action Movie Actor - Real Steel

Nominations[link]

• 1997 Mo Award for Best Actor in a Musical – Beauty and the Beast
• 1998 Olivier Award for Best Actor in a Musical – Oklahoma!
• 1999 Australian Film Institute for Best Performance by an Actor in a Leading Role - Erskineville Kings
• 2001 Golden Globe for Best Actor in a Motion Picture Musical or Comedy – Kate & Leopold
• 2001 CFCA Award for Most Promising Actor
• 2001 Blockbuster Entertainment Awards for Favorite Male - X-Men
• 2001 MTV Movie Awards for Best On-Screen Team - X-Men
• 2001 MTV Movie Awards for Breakthrough Male Performance - X-Men
• 2002 Golden Globes for Best Performance by an Actor in a Motion Picture - Kate & Leopold
• 2003 Teen Choice Award for Choice Movie Actor - X-Men 2
• 2004 Empire Awards for Best Actor - X-Men 2
• 2004 MTV Movie Awards for Best Fight - X-Men 2
• 2004 Teen Choice Awards for Choice Movie Actor - Van Helsing
• 2004 Teen Choice Awards for Choice Movie Chemistry - Van Helsing
• 2005 People's Choice Awards for Favorite Male Action Movie Star
• 2005 Emmy Award for Outstanding Individual Performance in a Variety or Music Program - [[58th Tony Awards|58th Annual Tony Awards Ceremonies
• 2006 Emmy Award for Outstanding Individual Performance in a Variety or Music Program – 59th Annual Tony Awards Ceremonies
• 2006 Green Room Award for Best Male Artist in a Leading Role – The Boy from Oz
• 2006 Teen Choice Awards for Choice Actor - X-Men: The Last Stand
• 2006 Teen Choice Awards for Choice Liplock - X-Men: The Last Stand
• 2007 Academy of Science Fiction, Fantasy & Horror Films for Best Actor - The Fountain
• 2007 Australian Film Institute for Best Actor - The Prestige
• 2009 Emmy Award for Outstanding Special Class Programs - 81st Academy Awards
• 2009 Teen Choice Awards for Choice Movie Drama Actor - Australia
• 2009 Teen Choice Awards for Choice Movie Hissy Fit - X-Men Origins: Wolverine
• 2009 Teen Choice Awards for Choice Movie Rumble - X-Men Origins: Wolverine
• 2010 MTV Movie Awards for Best Fight - X-Men Origins: Wolverine
• 2010 People's Choice Awards for Favorite Movie Actor
• 2010 People's Choice Awards for Favorite On-Screen Team - X-Men Origins: Wolverine
• 2010 SFX Awards for Best Actor - X-Men Origins: Wolverine
• 2012 People's Choice Awards for Favorite Movie Actor

Mentions in popular culture[link]

• Jackman was chosen as People magazine's Sexiest Man Alive of 2008.^[2]
• In the ABC comedy-drama Scrubs, one of the more prominent running gags is Dr. Cox's (John C. McGinley) seemingly irrational hatred of Jackman.
• In the 7th season of Punk'd, Jackman was led to believe that he had accidentally blown up director Brett Ratner's house.^[69]
• On the Season 6, Episode 13 of Will & Grace, which aired on 10 February 2004, the character Jack McFarland (Sean Hayes), mentions he is going to see The Boy From Oz, because he can not wait to see Hugh Jackman.^[70] He later discusses wanting to sue Jackman for stealing some of his dance moves.

References[link]