A symbol of the set of real numbers (ℝ)

Real numbers can be thought of as points on an infinitely long number line.

In mathematics, a real number is a value that represents a quantity along a continuous line. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers such as √2 (1.41421356... the square root of two, an irrational algebraic number) and π (3.14159265..., a transcendental number). Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one. The real line can be thought of as a part of the complex plane, and correspondingly, complex numbers include real numbers as a special case.

These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers — indeed, the realization that a better definition was needed — was one of the most important developments of 19th century mathematics. The currently standard axiomatic definition is that real numbers form the unique complete totally ordered field (R,+,·,<), up to isomorphism,^[1] Whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, or certain infinite "decimal representations", together with precise interpretations for the arithmetic operations and the order relation. These definitions are equivalent in the realm of classical mathematics.

Basic properties[link]

A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero. Real numbers are used to measure continuous quantities. They may in theory be expressed by decimal representations that have an infinite sequence of digits to the right of the decimal point; these are often represented in the same form as 324.823122147… The ellipsis (three dots) indicate that there would still be more digits to come.

More formally, real numbers have the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that if a nonempty set of real numbers has an upper bound, then it has a least upper bound. The second condition distinguishes the real numbers from the rational numbers: for example, the set of rational numbers whose square is less than 2 is a set with an upper bound (e.g. 1.5) but no least upper bound: hence the rational numbers do not satisfy the least upper bound property.

In physics[link]

In the physical sciences, most physical constants such as the universal gravitational constant, and physical variables, such as position, mass, speed, and electric charge, are modeled using real numbers. In fact, the fundamental physical theories such as classical mechanics, electromagnetism, quantum mechanics, general relativity and the standard model are described using mathematical structures, typically smooth manifolds or Hilbert spaces, that are based on the real numbers although actual measurements of physical quantities are of finite accuracy and precision.

In some recent developments of theoretical physics stemming from the holographic principle, the Universe is seen fundamentally as an information store, essentially zeroes and ones, organized in much less geometrical fashion and manifesting itself as space-time and particle fields only on a more superficial level. This approach removes the real number system from its foundational role in physics and even prohibits the existence of infinite precision real numbers in the physical universe by considerations based on the Bekenstein bound.^[2]

In computation[link]

Computer arithmetic cannot directly operate on real numbers, but only on a finite subset of rational numbers, limited by the number of bits used to store them, whether as floating-point numbers or arbitrary precision numbers. However, computer algebra systems can operate on irrational quantities exactly by manipulating formulas for them (such as Failed to parse (Missing texvc executable; please see math/README to configure.): \textstyle\sqrt 2 , Failed to parse (Missing texvc executable; please see math/README to configure.): \textstyle\arcsin \left({{2}\over{23}}\right) , orFailed to parse (Missing texvc executable; please see math/README to configure.): \textstyle\int_{0}^{1} {x^{x}}\;dx ) rather than their rational or decimal approximation;^[3] however, it is not in general possible to determine whether two such expressions are equal (the Constant problem).

A real number is called computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms,^[4] but an uncountable number of reals, almost all real numbers fail to be computable. Moreover, the equality of two computable numbers is an undecidable problem. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable.

Notation[link]

Mathematicians use the symbol R (or alternatively, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R} , the letter "R" in blackboard bold, Unicode ℝ – U+211D) to represent the set of all real numbers (as this set is naturally endowed with a structure of field, the expression field of the real numbers is more frequently used than set of all real numbers). The notation Rⁿ refers to the Cartesian product of n copies of R, which is an n-dimensional vector space over the field of the real numbers; this vector space may be identified to the n-dimensional space of Euclidean geometry as soon as a coordinate system has been chosen in the latter. For example, a value from R³ consists of three real numbers and specifies the coordinates of a point in 3-dimensional space.

In mathematics, real is used as an adjective, meaning that the underlying field is the field of the real numbers (or the real field). For example real matrix, real polynomial and real Lie algebra. As a substantive, the term is used almost strictly in reference to the real numbers themselves (e.g., The "set of all reals").

History[link]

Vulgar fractions had been used by the Egyptians around 1000 BC; the Vedic "Sulba Sutras" ("The rules of chords") in, ca. 600 BC, include what may be the first 'use' of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematicians since Manava (c. 750–690 BC), who were aware that the square roots of certain numbers such as 2 and 61 could not be exactly determined.^{[5][verification needed]} Around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2.

The Middle Ages brought the acceptance of zero, negative, integral, and fractional numbers, first by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects,^[6] which was made possible by the development of algebra. Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers.^[7] The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation, often in the form of square roots, cube roots and fourth roots.^{[8][verification needed]}

In the 16th century, Simon Stevin created the basis for modern decimal notation, and insisted that there is no difference between rational and irrational numbers in this regard.

In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones.

In the 18th and 19th centuries there was much work on irrational and transcendental numbers. Johann Heinrich Lambert (1761) gave the first flawed proof that π cannot be rational;^{[citation needed]} Adrien-Marie Legendre (1794) completed the proof, and showed that π is not the square root of a rational number. Paolo Ruffini (1799) and Niels Henrik Abel (1842) both constructed proofs of Abel–Ruffini theorem: that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots.

Évariste Galois (1832) developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Joseph Liouville (1840) showed that neither e nor e² can be a root of an integer quadratic equation, and then established existence of transcendental numbers, the proof being subsequently displaced by Georg Cantor (1873). Charles Hermite (1873) first proved that e is transcendental, and Ferdinand von Lindemann (1882), showed that π is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and has finally been made elementary by Adolf Hurwitz and Paul Gordan.

The development of calculus in the 18th century used the entire set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871. In 1874 he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, which he published in 1891. See Cantor's first uncountability proof.

Definition[link]

The real number system Failed to parse (Missing texvc executable; please see math/README to configure.): (\mathbb R,+,\cdot,<)

can be defined axiomatically up to an isomorphism, which is described below. There are also many ways to construct "the" real number system, for example, starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers. Another possibility is to start from some rigorous axiomatization of Euclidean geometry (Hilbert, Tarski etc.) and then define the real number system geometrically. From the structuralist point of view all these constructions are on equal footing.

Axiomatic approach[link]

Let R denote the set of all real numbers. Then:

• The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
• The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z:
  • if x ≥ y then x + z ≥ y + z;
  • if x ≥ 0 and y ≥ 0 then xy ≥ 0.
• The order is Dedekind-complete; that is, every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R.

The last property is what differentiates the reals from the rationals. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational.

The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind-complete ordered fields R₁ and R₂, there exists a unique field isomorphism from R₁ to R₂, allowing us to think of them as essentially the same mathematical object.

For another axiomatization of R, see Tarski's axiomatization of the reals.

Construction from the rational numbers[link]

The real numbers can be constructed as a completion of the rational numbers in such a way that a sequence defined by a decimal or binary expansion like (3, 3.1, 3.14, 3.141, 3.1415,...) converges to a unique real number. For details and other constructions of real numbers, see construction of the real numbers.

Properties[link]

Completeness[link]

A main reason for using real numbers is that the reals contain all limits. More precisely, every sequence of real numbers having the property that consecutive terms of the sequence become arbitrarily close to each other necessarily has the property that after some term in the sequence the remaining terms are arbitrarily close to some specific real number. In mathematical terminology, this means that the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). This is formally defined in the following way:

A sequence (x_n) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |x_n − x_m| is less than ε for all n and m that are both greater than N. In other words, a sequence is a Cauchy sequence if its elements x_n eventually come and remain arbitrarily close to each other.

A sequence (x_n) converges to the limit x if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |x_n − x| is less than ε provided that n is greater than N. In other words, a sequence has limit x if its elements eventually come and remain arbitrarily close to x.

Notice that every convergent sequence is a Cauchy sequence. The converse is also true:

Every Cauchy sequence of real numbers is convergent to a real number.

That is, the reals are complete.

Note that the rationals are not complete. For example, the sequence (1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...), where each term adds a digit of the decimal expansion of the positive square root of 2, is Cauchy but it does not converge to a rational number. (In the real numbers, in contrast, it converges to the positive square root of 2.)

The existence of limits of Cauchy sequences is what makes calculus work and is of great practical use. The standard numerical test to determine if a sequence has a limit is to test if it is a Cauchy sequence, as the limit is typically not known in advance.

For example, the standard series of the exponential function

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{e}^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}

converges to a real number because for every x the sums

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{n=N}^{M} \frac{x^n}{n!}

can be made arbitrarily small by choosing N sufficiently large. This proves that the sequence is Cauchy, so we know that the sequence converges even if the limit is not known in advance.

"The complete ordered field"[link]

The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.

First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element z, z + 1 is larger), so this is not the sense that is meant.

Additionally, an order can be Dedekind-complete, as defined in the section Axioms. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way.

These two notions of completeness ignore the field structure. However, an ordered group (in this case, the additive group of the field) defines a uniform structure, and uniform structures have a notion of completeness (topology); the description in the section Completeness above is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a characterisation of the real numbers.) It is not true that R is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field, and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Since it can be proved that any uniformly complete Archimedean field must also be Dedekind-complete (and vice versa, of course), this justifies using "the" in the phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way.

But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of R. Thus R is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.

Advanced properties[link]

The reals are uncountable, that is, there are strictly more real numbers than natural numbers, even though both sets are infinite. In fact, the cardinality of the reals equals that of the set of subsets (i.e., the power set) of the natural numbers, and Cantor's diagonal argument states that the latter set's cardinality is strictly bigger than the cardinality of N. Since only a countable set of real numbers can be algebraic, almost all real numbers are transcendental. The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis. The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms of set theory.

As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.

The real numbers form a metric space: the distance between x and y is defined as the absolute value |x − y|. By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology – in the order topology as intervals, in the metric topology as epsilon-balls. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. The reals are a contractible (hence connected and simply connected), separable and complete metric space of Hausdorff dimension 1. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals.

Every nonnegative real number has a square root in R, although no negative number does. This shows that the order on R is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make R the premier example of a real closed field. Proving this is the first half of one proof of the fundamental theorem of algebra.

The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalised such that the unit interval [0,1] has measure 1.

The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the Löwenheim–Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first order logic as the real numbers themselves. The set of hyperreal numbers satisfies the same first order sentences as R. Ordered fields that satisfy the same first-order sentences as R are called nonstandard models of R. This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it in R), we know that the same statement must also be true of R.

Generalizations and extensions[link]

The real numbers can be generalized and extended in several different directions:

• The complex numbers contain solutions to all polynomial equations and hence are an algebraically closed field unlike the real numbers. However, the complex numbers are not an ordered field.
• The affinely extended real number system adds two elements +∞ and −∞. It is a compact space. It is no longer a field, not even an additive group, but it still has a total order; moreover, it is a complete lattice.
• The real projective line adds only one value ∞. It is also a compact space. Again, it is no longer a field, not even an additive group. However, it allows division of a non-zero element by zero. It is not ordered anymore.
• The long real line pastes together ℵ₁* + ℵ₁ copies of the real line plus a single point (here ℵ₁* denotes the reversed ordering of ℵ₁) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of ℵ₁ in the long real line but not in the real numbers. The long real line is the largest ordered set that is complete and locally Archimedean. As with the previous two examples, this set is no longer a field or additive group.
• Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and thus are not Archimedean.
• Self-adjoint operators on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra. Positive-definite operators correspond to the positive reals and normal operators correspond to the complex numbers.

"Reals" in set theory[link]

In set theory, specifically descriptive set theory, the Baire space is used as a surrogate for the real numbers since the latter have some topological properties (connectedness) that are a technical inconvenience. Elements of Baire space are referred to as "reals".

Real numbers and logic[link]

The real numbers are most often formalized using the Zermelo-Fraenkel axiomatization of set theory, but some mathematicians study the real numbers with other logical foundations of mathematics. In particular, the real numbers are also studied in reverse mathematics and in constructive mathematics.^[9]

Abraham Robinson's theory of nonstandard or hyperreal numbers extends the set of the real numbers by infinitesimal numbers, which allows building infinitesimal calculus in a way closer to the usual intuition of the notion of limit. Edward Nelson's internal set theory is a non-Zermelo-Fraenkel set theory that considers non-standard real numbers as elements of the set of the reals (and not of an extension of it, as in Robinson's theory).

The continuum hypothesis posits that the cardinality of the set of the real numbers is Failed to parse (Missing texvc executable; please see math/README to configure.): \aleph_1 , i.e. the smallest infinite cardinal number after Failed to parse (Missing texvc executable; please see math/README to configure.): \aleph_0 , the cardinality of the integers. Paul Cohen proved in 1963 that it is an axiom independent of the other axioms of set theory; that is, one may choose either the continuum hypothesis or its negation as an axiom of set theory, without contradiction.

See also[link]

Notes[link]

References[link]

• Georg Cantor, 1874, "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen", Journal für die Reine und Angewandte Mathematik, volume 77, pages 258–262.
• Robert Katz, 1964, Axiomatic Analysis, D. C. Heath and Company.
• Edmund Landau, 2001, ISBN 0-8218-2693-X, Foundations of Analysis, American Mathematical Society.
• Howie, John M., Real Analysis, Springer, 2005, ISBN 1-85233-314-6

External links[link]

• The real numbers: Pythagoras to Stevin
• The real numbers: Stevin to Hilbert
• The real numbers: Attempts to understand
• What are the "real numbers," really?

Justin Bieber
Justin Bieber at the NRJ Music Awards in Cannes, France in 2012
Background information
Birth name	Justin Drew Bieber[1]
Born	(1994-03-01) March 1, 1994 (age 18)[2] London, Ontario, Canada
Origin	Stratford, Ontario, Canada
Genres	Pop, R&B, teen pop[3][4][5]
Occupations	Singer-songwriter, musician, actor
Instruments	Vocals, guitar, piano, percussion,[6] trumpet[7]
Years active	2008–present
Labels	Island, RBMG
Associated acts	Usher, Ludacris
Website	justinbiebermusic.com

Justin Drew Bieber ( /ˈbiːbər/ BEE-bər, born March 1, 1994)^[2] is a Canadian Pop/R&B singer-songwriter, musician, and actor.^[3][5] Bieber was discovered in 2008 by Scooter Braun,^[8] who came across Bieber's videos on YouTube and later became his manager. Braun arranged for him to meet with Usher in Atlanta, Georgia, and Bieber was soon signed to Raymond Braun Media Group (RBMG), a joint venture between Braun and Usher,^[9] and then to a recording contract with Island Records offered by L.A. Reid.^[6][10] Bieber's debut single, "One Time", was released in 2009 and peaked in the top twenty in Canada and charted in the top thirty in several international markets. His debut album, the seven-track EP My World, followed in November 2009, and was soon certified platinum in the United States. He became the first artist to have seven songs from a debut album chart on the Billboard Hot 100.^[11]

Bieber's first full-length studio album, My World 2.0, was released in March 2010. It debuted at number one or within the top ten in several countries and was certified platinum in the United States. It was preceded by the worldwide top-ten single, "Baby". The music video of "Baby" is currently ranked as the most viewed and most discussed YouTube video.^[12] Bieber followed up the release of his debut album with his first headlining tour, the My World Tour, the remix albums My Worlds Acoustic and Never Say Never – The Remixes, and the 3D biopic-concert film Justin Bieber: Never Say Never, that nearly matched the record for the biggest opening weekend gross for a concert-film. Bieber released his second studio album, Under the Mistletoe, in November 2011, which debuted at number one on the Billboard 200. Bieber has received numerous awards, including Artist of the Year at the 2010 American Music Awards, and has been nominated for such awards as Best New Artist and Best Pop Vocal Album at the 53rd Grammy Awards. In 2012, Bieber was named Forbes 3rd most powerful celebrity in the world, earning $55 million. As of May 2012, Bieber has sold 15 million albums.^[13]

Early life[link]

Bieber was born on March 1, 1994, in London, Ontario in St Joseph's Hospital,^[14][15] and was raised in Stratford, Ontario.^[16] Bieber's mother, Patricia Lynn "Pattie" Mallette, was 18 years old when she became pregnant. Mallette, originally from Stratford, raised her son with the help of her parents, Bruce and Diane.^[17] Mallette worked a series of low-paying office jobs, raising Bieber as a single mother in low-income housing. Bieber has maintained contact with his father, Jeremy Jack Bieber, who married another woman and had two children.^[18][19][20] Bieber's great-grandfather was from Germany.^[21] Bieber's mother's ancestry is French Canadian.^[22] Bieber attended a French immersion elementary school in Stratford called Jeanne Sauvé Catholic School.^[17]

During his childhood, Bieber was interested in hockey, soccer, and chess; he often kept his musical aspirations to himself.^[18] As he grew up, Bieber taught himself to play the piano, drums, guitar, and trumpet.^[16][23] In early 2007, when he was twelve, Bieber sang Ne-Yo's "So Sick" for a local singing competition in Stratford and was placed second.^[6][23] Mallette posted a video of the performance on YouTube for their family and friends to see. She continued to upload videos of Bieber singing covers of various R&B songs, and Bieber's popularity on the site grew.^[10] Chris Hicks, Bieber's A&R at Island/Def Jam, explained the young artist's huge online following to HitQuarters by saying:

"He was doing something different. He was an attractive white kid singing very soulful R&B hits. That set him apart immediately from anyone in his range because no one was covering or singing these kind of records. But equally important was that you believed in these songs – it was real. And you wanted to hear more."^[24]

Career[link]

[edit] 2008–09: Discovery and My World

While searching for videos of a different singer, Scooter Braun, a former marketing executive of So So Def, clicked on one of Bieber's 2007 videos by accident.^[18] Impressed, Braun tracked down the theater Bieber was performing in, located Bieber's school, and finally contacted Mallette. Mallette was reluctant because of Braun's Jewish religion; she remembered praying, "God, I gave him to you. You could send me a Christian man, a Christian label! ... you don’t want this Jewish kid to be Justin’s man, do you?"^[18] However, after praying with her church elders and receiving their encouragement, she permitted Braun to fly Bieber, then 13,^[10] to Atlanta, Georgia, to record demo tapes.^[18] A week after arriving, Bieber sang for R&B singer/songwriter Usher.^[25] Bieber was soon signed to Raymond Braun Media Group (RBMG), a joint venture between Braun and Usher.^[9] Justin Timberlake was also reportedly in the running to sign Bieber, but lost the bidding war to Usher.^[9][26] Usher then sought assistance in finding a label home for the artist from then manager Chris Hicks, who helped engineer an audition with his contact Antonio L.A. Reid of Island Def Jam Music Group.^[24] Reid signed Bieber to Island Records in October 2008 (resulting in a joint venture between RBMG and Island Records) and appointed Hicks as executive Vice-President of Def Jam where he would be able manage Bieber's career at the label.^{[6][9][10][24]} At that point, Bieber and his mother moved to Atlanta temporarily, also the home of Usher and Braun, to record and get counseling from Braun.^[9] Braun became Bieber's manager.^[10]

Bieber in September 2009 at the Nintendo World Store

Bieber's first single, "One Time", was released to radio while Bieber was still recording his debut album.^[27] The song reached number 12 on the Canadian Hot 100 during its first week of release in July 2009,^[10] and later peaked at number 17 on the Billboard Hot 100. During fall 2009 it had success in international markets.^[28] The song was certified Platinum in Canada and the US and Gold in Australia and New Zealand.^[29][30][31] Bieber's first release, an extended play entitled My World, was released on November 17, 2009. The album's second single, "One Less Lonely Girl", and two promo singles, "Love Me", and "Favorite Girl", were released exclusively on the iTunes store and charted within the top forty of the Billboard Hot 100.^[28] "One Less Lonely Girl" was later also released to radio and peaked within the top fifteen in Canada and the US, being certified Gold in the latter.^[28] My World was eventually certified Platinum in the US and Double Platinum in both Canada and the UK.^[31][32][33] To promote the album, Bieber performed on several live shows such as mtvU's VMA 09 Tour, European program The Dome, YTV's The Next Star, The Today Show,^[34][35] The Wendy Williams Show, Lopez Tonight, The Ellen DeGeneres Show, It's On with Alexa Chung, Good Morning America, Chelsea Lately, and BET's 106 & Park with Rihanna. Bieber also guest starred in an episode of True Jackson, VP in late 2009.^[36]

Bieber performed Stevie Wonder's "Someday at Christmas" for U.S. President Barack Obama and first lady Michelle Obama at the White House for Christmas in Washington, which was broadcast on December 20, 2009, on U.S. television broadcaster TNT.^[37] Bieber was also one of the performers at Dick Clark's New Year's Rockin' Eve with Ryan Seacrest on December 31, 2009.^[38] Bieber was a presenter at the 52nd Grammy Awards on January 31, 2010. He was invited to be a vocalist for the remake of We Are The World (a song written by Michael Jackson and Lionel Richie) for its 25th anniversary to benefit Haiti after the earthquake. Bieber sings the opening line, which was sung by Lionel Richie in the original version.^[39] On March 12, 2010, a version of K'naan's "Wavin' Flag" recorded by a collective of Canadian musicians known as Young Artists for Haiti was released. Bieber is featured in the song, performing the closing lines.

[edit] 2010–11: My World 2.0 and Never Say Never film

"Baby", the lead single from his debut album, My World 2.0, which features Ludacris, was released in January 2010 and became his biggest hit thus far, charting at number five in the U.S. and reaching the top ten in seven other countries.^[28] Two promo singles "Never Let You Go", and "U Smile" were top thirty hits on the U.S. Hot 100, and top twenty hits in Canada.^[28] According to review aggregator Metacritic, the album has received "generally favorable reviews".^[40] It debuted at number one on the U.S. Billboard 200, making Bieber the youngest solo male act to top the chart since Stevie Wonder in 1963.^[41] My World 2.0 also debuted at number one on the Canadian Albums Chart, Irish Albums Chart, Australian Albums Chart, and the New Zealand Albums Chart^[28] and reached the top ten of fifteen other countries.^[42][43] To promote the album, Bieber appeared on several live programs including The View, the 2010 Kids Choice Awards, Nightline, The Late Show with David Letterman, The Dome and 106 and Park.^[44] Bieber collaborated with Sean Kingston on his single "Eenie Meenie" which also appeared on Bieber's debut album. The song reached the top ten in the United Kingdom and Australia, and the top-twenty of most other markets.

Bieber performing at the Conseco Fieldhouse, in August 2010

On April 10, 2010, Bieber was the musical guest on Saturday Night Live.^[45] On July 4, 2010, Bieber performed at the Macy's Fourth of July Fireworks Spectacular in New York City.^[46] The second single from My World 2.0, "Somebody to Love", was released in April 2010, and a remix was released featuring Bieber's mentor Usher. The single reached number ten in Canada, number fifteen in the United States, and the top twenty in multiple international countries.

On June 23, 2010, Bieber went on his first official headlining tour, the My World Tour, starting in Hartford, Connecticut, to promote My World and My World 2.0.^[47] In July 2010, it was reported that Bieber was the most searched for celebrity on the Internet.^[48] That same month his music video, "Baby", surpassed Lady Gaga's "Bad Romance" to become the most viewed, and most disliked YouTube video ever.^[49] In September 2010, it was reported that Bieber accounted for three percent of all traffic on Twitter, according to an employee of the social-networking site.^[50]

Bieber began recording his second album in July 2010 in New York City.^[51] At this point, because of puberty, his voice was deeper than it was when he recorded his first album. In April 2010, the singer discussing his vocals remarked, "It cracks. Like every teenage boy, I'm dealing with it and I have the best vocal coach in the world. [...] Some of the notes I hit on "Baby" I can't hit anymore. We have to lower the key when I sing live."^[52]

Bieber performing in Jakarta during his My World Tour.

Bieber guest-starred in the season premiere of the CBS crime drama CSI: Crime Scene Investigation, which aired on September 23, 2010. He played a "troubled teen who is faced with a difficult decision regarding his only brother", who is also a serial bomber. Bieber was also in a subsequent episode which aired on February 17, 2011, in which his character is killed.^[53] Bieber performed a medley of "U Smile", "Baby", and "Somebody to Love" and briefly played the drums at the 2010 MTV Video Music Awards on September 12, 2010.^[54] Bieber announced in October 2010 that he would release an acoustic album, called My Worlds Acoustic.^[55] It was released on Black Friday in the United States and featured acoustic versions of songs from his previous albums, and accompanied the release of a completely new song titled "Pray".^[56]

A 3-D part-biopic, part-concert film starring Bieber entitled Justin Bieber: Never Say Never, was released on February 11, 2011, directed by Step Up 3D director Jon Chu. It topped the box office with an estimated gross of $12.4 million on its opening day from 3,105 theaters.^[57] It grossed $30.3 million for the weekend and was narrowly beaten by the romantic comedy Just Go With It, which grossed $31 million.^[58] Never Say Never is said to have exceeded industry expectations, nearly matching the $31.1 million grossed by Miley Cyrus's 2008 3-D concert film, Hannah Montana & Miley Cyrus: Best of Both Worlds Concert, which holds the record for the top debut for a music-documentary.^[59][60] Never Say Never grossed a total of $98,441,954 worldwide.^[61] The film is accompanied by his second remix album, Never Say Never – The Remixes, released February 14, 2011, and features remixes of songs from his debut album, with guest appearances from Miley Cyrus, Chris Brown, and Kanye West, among others.^[62]

[edit] 2011–present: Under The Mistletoe and Believe

On February 27, 2011, Bieber attended the 2011 Vanity Fair Oscar Party with American actress and singer Selena Gomez,^[63] confirming several months of media speculation about a romantic relationship between the pair.^{[64][65][66][67][68][69]}

In June 2011, Bieber was ranked number 2 on the Forbes list of Best-Paid Celebrities under 30. He is the youngest star and 1 of 7 musicians on the list raking in with $53 million earned in a 12 month period.^[70] On November 1, 2011, Bieber released Under the Mistletoe, his second studio album.^[71] It debuted at number one on the Billboard 200, selling 210,000 copies in its first week of release.^[72]

In late 2011, Bieber began recording his third studio album, entitled Believe. Kanye West and Drake are confirmed to be featured on the album.^[73] On February 22, 2012, Bieber announced via Twitter that the first single off Believe will be released in March 2012.^[74] The following week, Bieber appeared on The Ellen DeGeneres Show to announce that the first single will be called "Boyfriend" and was released on March 26, 2012.^[75] The song was co-written by Mike Posner.^[76] The second single, Die In Your Arms, was released on May 29th, 2012. ^[77]

Image[link]

Justin Bieber has a higher score on the Klout online "influence" scale than Barack Obama or The Dalai Lama.^[78] According to Jan Hoffman of The New York Times, part of Bieber's appeal stems from his YouTube channel. Long before he released his EP, My World, in mid-November, the YouTube videos attracted millions of views.^[18] Braun recognized the appeal. Prior to flying him to Atlanta, Braun wanted to "build him up more on YouTube first" and had Bieber record more home videos for the channel. "I said: 'Justin, sing like there’s no one in the room. But let's not use expensive cameras.' We'll give it to kids, let them do the work, so that they feel like it's theirs", recalled Braun.^[18] Bieber continues to upload videos to the same channel and has opened a Twitter account, from which he interacts with fans regularly;^[10] his account was reported in November 2010 to have over 6 million followers.^[79] Since then he has been consistently gaining followers at an average of 24,000 per day.^[80] The accounts also serve marketing purposes; for example, Bieber's music video for "One Time" only began selling quickly after it was uploaded to YouTube.^[81]

Bieber performing "Favorite Girl" in Zurich, Switzerland April 2011.

Usher comments that while he and Bieber were both signed at the same age, "I had the chance to ramp up my success, where this has happened to Bieber abruptly." As a result, Usher, Braun, Bieber's bodyguard Kenny, and other adults surrounding Bieber constantly coach him on handling fame and his public image.^[18] After signing Bieber, Usher appointed one of his former assistants, Ryan Good, to be Bieber's road manager and stylist. Good, once nicknamed Bieber's "swagger coach", created a "streetwise look" for the singer which consisted of baseball caps, hoodies, dog chains and flashy sneakers. Amy Kaufman of The Los Angeles Times comments, "Though a product of a middle-class suburban upbringing in Stratford, Ontario, Bieber's manner of dress and speech ("Wassup man, how you doin'?" or "It's like, you know, whateva' ") suggest he's mimicking his favorite rappers."^[82]

Bieber is often featured in teen magazines such as Tiger Beat,^[18] and has been labeled as a "teen heartthrob".^[83] Bieber has released a collection of nail varnishes to raise awareness for charity.^[84] Wax statues of Bieber are on display at Madame Tussauds wax museums in New York, Amsterdam and London.^[85] His change of hairstyle in 2010, and the consequent alterations to Bieber products, led to it being called 'the most expensive musical haircut of all time;^[86] one company spent $100,000 to fix its dolls for the 2011 Christmas season.^[87]

Bieber has been criticized for looking and sounding younger than his age, his teen-pop music, image, and frequent media attention.^[88] He has been a frequent target of Internet bloggers and message board posters, notably by users of Internet message board 4chan and users of YouTube.^[89] Nick Collins of The Daily Telegraph speculated that "Bieber's character also appears to strike a particularly sour note with his Internet critics, with many remarks commenting on his youthful appearance, his teen-pop songs, his image as a heart-throb to young teenage girls and his manner of speech".^[90]

Personal views and life[link]

Bieber is a Christian,^[91] and said he has a relationship with Jesus, talks to him and that "he's the reason I'm here".^[92]

Bieber's comments in a February 2011 profile in Rolling Stone sparked controversy.^[93] Asked about abstinence, Bieber responded, "I don't think you should have sex with anyone unless you love them." He said he does not "believe in abortion" and that it is "like killing a baby".^[94] He described sexual orientation as "everyone's own decision."^[95] Bieber has contributed to the It Gets Better Project,^[96] which aims to prevent suicide among LGBT youth.^[97]

Bieber has said he is not interested in obtaining United States citizenship and has criticized America's health care system. Praising Canada as being "the best country in the world", he cited its health care system as a model example.^[94]

In May 2012, Bieber purchased a 10,000 square-foot mansion on 1.3 acres of land, located in the community of Calabasas, California, a suburb of Los Angeles.^[98]

Twitter[link]

As of April 2012^[update], Bieber's Twitter account (@justinbieber) is the second most popular celebrity account behind that of Lady Gaga.^[99] By mid-April 2012, he had over 20 million followers.^[100][101] His follower acquisition is one new follower every other second.^[102][103] His popularity on Twitter at one point accounted for three percent of all Twitter related traffic, with a Twitter employee commenting that "racks of servers are dedicated" to Bieber.^{[103][104][105][106][107][108][109][110][111][112]} This resulted in over 180 million page views for the service each month.^[105] Bieber was frequently a trending topic on Twitter when the feature first launched^[105] because his fans frequently discussed him on the network. ^[113] and was named the top trending star on Twitter in 2010;^[114] Klout gave his account a score of 100.^[115][116]

The size of Bieber's and Gaga's follower bases are cited as a reason why marketers should pay attention to Twitter:^[117] with an ability to reach millions of people with a single tweet.^[117] 8.3% of tweets mentioning "bieber" were semi-automated and probably Twitter related spam.^[118] Research done about Twitter and the 2011 Egyptian revolution includes Bieber as he made tweets about the topic at a time when he had roughly 8 million followers. His multiple tweets resulted in 32,000 responses each, which made Bieber's Twitter account one of the single largest nodes for discussion about the uprising one of the largest on Twitter.^[119][120]

In March 2012, Bieber fans launched an unsuccessful attempt to unseat Lady Gaga as the most popular celebrity on Twitter. Neither Bieber nor Lady Gaga discussed their mutual fanbase's campaigns on Twitter.^[121]

Discography[link]

Justin Bieber portal

• My World 2.0 (2010)
• Under the Mistletoe (2011)
• Believe (2012)

Tours[link]

• 2009: Urban Behavior Tour
• 2010–2011: My World Tour
• 2012-2013: Believe Tour

Filmography[link]

Awards and nominations[link]

References[link]