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Complex number

2012-03-16T21:23:01Z

Nsmith: /* Applications */ Unbacked opinionated statement

{{Math-m}}

The '''complex numbers''' are a set of numbers which have important applications in the analysis of periodic, oscillatory, or wavelike phenomena. Mathematicians denote the set of complex numbers with an ornate capital letter: <math>\mathbb{C}</math>. They are the 5th item in this hierarchy of types of [[number]]s:

*The "[[natural number]]s", 1, 2, 3, ... (There is controversy about whether zero should be included. It doesn't matter.)
*The "[[integer]]s"—positive, negative, and [[zero]]
*The "[[rational number]]s", or [[fraction]]s, like 355/113
*The "[[real number]]s", including irrational numbers
*The "complex numbers", which give solutions to polynomial equations

A complex number is composed of two parts, or components—a ''real'' component and an ''imaginary'' component. Each of these components is an ordinary (that is, real) number. The complex numbers form an "extension" of the real numbers: If the imaginary component of a complex number is zero, that number is essentially identical to the real number that is its real component.

__TOC__


The complex numbers are defined as a 2-dimensional vector space over the [[real number]]s. That is, a complex number is an ordered pair of numbers: (a, b). The familiar real numbers constitute the complex numbers with second component zero. That is, x corresponds to (x, 0).

The second component is called the ''imaginary part''. Its unit basis vector is called <math>i</math>. The first component is called the ''real part''. Its unit basis vector is just 1. Thus, the complex number <math>(a, b)\,</math> can also be written <math>a + bi\,</math>. Numbers with real part of zero are sometimes called "pure imaginary", with the term "complex" reserved for numbers with both components nonzero.

While the "invisible" nature of the imaginary component may be disconcerting at first (and the word "imaginary" may be an unfortunate term for it), the complex numbers are just as genuine as the Dedekind cuts and Cauchy sequences that are used in the definition of the "real" numbers.

The complex numbers form a [[Field_(mathematics)|field]], with the mathematical operations defined as shown below.

==Arithmetic operations==

The field operations are defined as follows:

===Addition===
Addition is just the standard addition on the 2-dimensional vector space. That is, add the real parts (first components), and add the imaginary parts (second components).

Letting the complex numbers w and z be defined by their respective components:
:<math>w = (a, b) = a + bi\,</math>
:<math>z = (c, d) = c + di\,</math>
we have:
:<math>w + z = (a+c, b+d) = a+c + (b+d)i\,</math>

===Multiplication===
Multiplication has a special definition. It is this definition that gives the complex numbers their important properties.
:<math>w * z = (a*c - b*d) + (a*d + b*c)i\,</math>
:::When we multiply <math>(0+1i)*(0+1i)\,</math>, this definition gives <math>i^2 = -1\,</math>
Writing out the product <math>w * z\,</math> in the obvious way, we get the same answer:
:<math>(a + bi) * (c + di) = a*c + b*d*i^2 + (a*d + b*c)i = (a*c - b*d) + (a*d + b*c)i\,</math>
This means that, using <math>i^2 = -1\,</math>, one can perform arithmetic operations in a completely natural way.

===Division===
Division requires a special trick. We have:
:<math>\frac{w}{z} = \frac{a + bi}{c + di}\,</math>
To get the individual components, the denominator needs to be real. This can be accomplished by multiplying both numerator and denominator by <math>(c - di)\,</math>. We get:
:<math>\frac{w}{z} = \frac{(a + bi)*(c - di)}{(c + di)*(c - di)} = \frac{(a*c + b*d) + (b*c - a*d)i}{c^2 + d^2}\,</math>

This division will fail if and only if <math>c^2 + d^2 = 0\,</math>, that is, <math>c\,</math> and <math>d\,</math> are both zero, that is, the complex denominator <math>z\,</math> is exactly zero (both components zero). This is exactly analogous to the rule that real division fails if the denominator is exactly zero.

==Fundamental theorem of algebra==
The complex numbers form an [[algebraic closure|algebraically closed]] field. This means that any <math>n^{th}\,</math> degree polynomial can be factored into n first degree (linear) polynomials. Equivalently, such a polynomial has n roots (though one has to count all multiple occurrences of repeated roots.) This statement is the Fundamental Theorem of Algebra, first proved by [[Gauss|Carl Friedrich Gauss]] around 1800. The theorem is ''not true'' if the roots are required to be real. (This failure is what led to the development of complex numbers in the first place.) But when the roots are allowed to be complex, the theorem applies even to polynomials with complex coefficents.

The simplest polynomial with no real roots is <math>x^2 + 1\,</math>, since -1 has no real square root. But if we look for roots of the form <math>a + bi\,</math>, we have:
:<math>(a + bi)^2 + 1 = a^2 - b^2 + 1 + 2abi = 0\,</math>
For the imaginary part to be zero, one of a or b must be zero. For the real part to be zero, a must be zero and <math>b^2\,</math> must be 1. This means that <math>b = \pm 1\,</math>, so the roots are <math>(0, 1)\,</math> and <math>(0, -1)\,</math>, or <math>\pm{}i\,</math>.

These two numbers, <math>i\,</math> and <math>-i\,</math>, are the square roots of -1.

Similar analysis shows that, for example, 1 has three cube roots:
*<math>1\,</math>
*<math>-\frac{1}{2} + \frac{\sqrt{3}}{2}i\,</math>
*<math>-\frac{1}{2} - \frac{\sqrt{3}}{2}i\,</math>
One can verify that
<math>(x-1) \left(x + \frac{1}{2} - \frac{\sqrt{3}}{2}i\right) \left(x + \frac{1}{2} + \frac{\sqrt{3}}{2}i\right) = x^3 - 1\,</math>

For a given field, the field containing it (strictly speaking, the smallest such field) that is algebraically closed is called its algebraic closure. The field of real numbers is not algebraically closed; its closure is the field of complex numbers.

==Polar coordinates, modulus, and phase==
Complex numbers are often depicted in 2-dimensional Cartesian analytic geometry; this is called the ''complex plane''. The real part is the x-coordinate, and the imaginary part is the y-coordinate. When these points are analyzed in [[polar coordinates]], some very interesting properties become apparent. The representation of complex numbers in this way is called an ''Argand diagram''.

If a complex number is represented as <math>x + yi\,</math>, its polar coordinates are given by:
:<math>r = \sqrt{x^2 + y^2}\,</math>
:<math>\theta = \tan^{-1}\frac{y}{x}\,</math>
Transforming the other way, we have:
:<math>x = r \cos\theta\,</math>
:<math>y = r \sin\theta\,</math>

The radial distance from the origin, <math>r\,</math>, is called the ''modulus''. It is the complex equivalent of the absolute value for real numbers. It is zero if and only if the complex number is zero.

The angle, <math>\theta\,</math>, is called the ''phase''. (Some older books refer to it as the ''argument''.) It is zero for positive real numbers, and <math>\pi\,</math> radians (180 degrees) for negative ones.

The multiplication of complex numbers takes a particularly interesting and useful form when represented this way. If two complex numbers <math>z_1\,</math> and <math>z_2\,</math> are represented in modulus/phase form, we have:
:<math>z_1 = r_1 (\cos\theta_1 + i\sin\theta_1)\,</math>
:<math>z_2 = r_2 (\cos\theta_2 + i\sin\theta_2)\,</math>

:<math>z_1 * z_2 = r_1 * r_2 * ((\cos\theta_1 \cos\theta_2 - \sin\theta_1 \sin\theta_2) + i (\sin\theta_1 \cos\theta_2 + \cos\theta_1 \sin\theta_2))\,</math>

But that is just the addition rule for sines and cosines!

:<math>z_1 * z_2 = r_1 * r_2 * (\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2))\,</math>

So the rule for multiplying complex numbers on the Argand diagram is just:
:Multiply the moduli.
:Add the phases.

One can use this property to find all of the nth roots of a number geometrically. For example, by using these trigonometric formulas:
::<math>\sin(30) = \cos(60) = \frac{1}{2}\,</math>
::<math>\sin(60) = \cos(30) = \frac{\sqrt{3}}{2}\,</math>
The three cube roots of 1, listed above, can all be seen to have moduli of 1 and phases of 0 degrees, 120 degrees, and 240 degrees respectively. When raised to the third power, the phases are tripled, obtaining 0, 360, and 720 degrees. But they are all the same angle—zero. So the cubes of these numbers are all just one. Similarly, <math>i\,</math> and <math>-i\,</math> have phases of 90 degrees and 270 degrees. When those numbers are squared, the phases are 180 and 540, both of which are the same angle—180 degrees. So their squares are both -1.

One can apply this property to the nth roots of any complex number. They lie equally spaced on a circle. This is DeMoivre's theorem.

[[File:Argandcbrt1.jpg|thumb|left|400px|The three cube roots of +1.]]
[[File:Argandsqrtm1.jpg|thumb|right|400px|The two square roots of -1.]]
{{clear}}

===Euler's formula===

A careful analysis of the power series for the exponential, sine, and cosine functions reveals the marvelous ''Euler formula'':
:<math>z = r e^{i\theta}\,</math>
of which there is the famous case (for θ = π):
:<math>e^{i\pi} = -1\,</math>

==Conjugates==
The ''complex conjugate'', or just ''conjugate'', of a complex number is the result of negating its imaginary part. The conjugate is written with a bar over the quantity: <math>\overline{z}\,</math>. All real numbers are their own conjugates.

All arithmetic operations work naturally with conjugates—the sum of the conjugates is the conjugate of the sum, and so on.
:<math>\overline{w+z} = \overline{w} + \overline{z}\,</math>
:<math>\overline{w*z} = \overline{w} * \overline{z}\,</math>
:<math>\overline{w/z} = \overline{w} / \overline{z}\,</math>
It follows that, if P is a polynomial with real coefficients (so that its coefficients are their own conjugates)
:<math>\overline{P(z)} = P(\overline{z})\,</math>
If <math>z\,</math> is a root of a real polynomial, then, since zero is its own conjugate, <math>\overline{z}\,</math> is also a root. This is often expressed as "Non-real roots of real polynomials come in conjugate pairs." We saw that above for the cube roots of 1—two of the roots are complex and are conjugates of each other. The third root is its own conjugate.

==Transcendental functions==
The higher mathematical functions (often called "transcendental functions"), like exponential, log, sine, cosine, etc., can be defined in terms of power series ([[Taylor series]]). They can be extended to handle complex arguments in the completely natural way, so these functions are defined over the complex plane. They are in fact "complex [[analytic function]]s". Just about any normal function one can think of can be extended to the complex numbers, and is complex analytic. Since the power series coefficients of the common functions are real, they work naturally with conjugates. For example:
:<math>\sin(\overline{z}) = \overline{\sin(z)}\,</math>
:<math>\log(\overline{z}) = \overline{\log(z)}\,</math>

==Complex functions==
The general study of functions that take a complex argument and return a complex result is an extremely rich and useful area of mathematics, known as [[complex analysis]]. When such a function is differentiable, it is called a ''[[analytic function|complex analytic function]]'', or just ''analytic function''.

==Applications==
Complex numbers are extremely important in many areas of pure and applied mathematics and physics. Basically, any description of oscillatory phenomena can benefit from a formulation in terms of complex numbers.

The applications include these:
*Theoretical mathematics:
::algebra (including finding roots of polynomials)
::linear algebra—vector spaces, inner products, Hermitian and unitary operators, Hilbert spaces, etc.
::eigenvalue/eigenvector problems
::number theory—This seems improbable, but it is true. The Riemann zeta function, and the Riemann hypothesis, are very important in number theory. For a long time, the only proofs of the prime number theorem used complex numbers. The Wiles/Taylor proof of [[Fermat's Last Theorem]] uses modular forms, which use complex numbers.
*Applied mathematics
::eigenvalue/eigenvector problems
::Fourier and Laplace transforms
::linear differential equations
::stability theory
*Electrical engineering
::alternating current circuit analysis
::filter design
::antenna design
*Theoretical physics
::quantum mechanics
::fundamental particle physics
::electromagnetic radiation
::optics

[[Category:Mathematics]]
[[Category:Complex analysis]]

France

2012-03-16T21:12:04Z

Nsmith: /* Political Conditions */

{{Country
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|name =''République Française''
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|flag =Flag of France.png
|arms =Armoirie France.png
|capital =Paris
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|government =Republic
|government-raw =
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|king =
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|monarch-raw =
|president =Nicolas Sarkozy
|president-raw =
|chancellor =
|chancellor-raw =
|pm =François Fillon
|pm-raw =
|area =260,558 sq mi
|pop =65,312,000
|pop-basis =2011
|gdp =$2.555 trillion (2010)
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|gdp-pc =$40 591 (2010)
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'''France''' (Official title: ''République Française'', the ''French Republic'') is a major country in Western [[Europe]]. One of the oldest countries in Europe, France has played a significant role in world events since the era of Cardinal Richelieu in the mid-17th century and beyond.

==Geography==
[[File:Harbour of Marseille France.jpg|thumb|left|120px|The Harbour of Marseille.]]
*Total area: 674,843 sq. km. (260,558 sq. mi.); largest Western European country, about the size of [[Texas]].
*Cities: Capital (and largest city)--[[Paris]]. Major cities--[[Marseille]], [[Lyon]], [[Toulouse]], [[Strasbourg]], [[Nice]], [[Rennes]], [[Lille]], [[Bordeaux]].
*Terrain: Varied.
*Climate: Temperate.
*Population : 63 136 180 (January 2011 INSEE est.) excluding overseas territories.

 
 

==History==
See [[History of France]]

==People==
Since prehistoric times, France has been a crossroads of trade, travel, and migration. Three basic European ethnic stocks--Celtic, Latin, and Teutonic (Frankish)--have blended over the centuries to make up its present population. France's birth rate was among the highest in Europe from 1945 until the late 1960s. Since then, its birth rate has fallen but remains higher than that of most other west European countries. Traditionally, France has had a high level of [[immigration]]. More than 1 million [[Muslims]] immigrated in the 1960s and early 1970s from [[North Africa]], especially [[Algeria]]; their numbers are now above 5 million (2004 figure).

===Education===
[[File:Pascal by Augustin Pajou Louvre PD WC.jpg|thumb|Sculpture of [[Blaise Pascal]] by Augustin Pajou, Le Louvre.]]
Education is free, beginning at age 3 (sometimes 2), and mandatory between ages 6 and 16. The [[public education]] system is highly centralized. Private schools are primarily Roman Catholic, with some of them still run by religious orders historically involved in the French educational system. Almost all schools welcome students from all denominations.

In France, teachers abide by the state-chosen educational program, with the science curriculum based on [[Charles Darwin|Charles Darwin’s]] [[Theory of Evolution]]. [[Creationism]] can be referenced for cultural and philosophical values.
[[Image:Gtr7564.jpg|left|150px|thumb|[[René Descartes]], French philosopher & scientist]]
====Higher education====
Higher education in France began with the founding of the [[University of Paris]] in 1150. It now consists of 91 public universities and 175 professional schools, including the post-graduate Grandes Ecoles. Private, college-level institutions focusing on business and management with curricula structured on the American system of credits and semesters have been growing in recent years.

France has an elaborately stratified system of higher education; nearly all schools are controlled and funded by the government. There are few private foundations or endowments, but some research money comes from American foundations. Most scholars and scientists at all levels are civil servants with lifetime tenure. Rigid hierarchies ensure that young researchers pay their dues before moving up the ladder to supervisory roles. Paris decides what areas of research are high priorities, which laboratories will be “centers of excellence,” and how many people will be hired in what fields. It also classifies France’s top educational and research institutions as ''grands organismes'', ''grands établissements'', and ''grandes écoles'', with bureaucratic strictures ensuring that there will be little contact between the different groups.<ref> Fred Schwarz, "The Uncertainty Principle: America dominates science with a classic American formula: freedom," ''National Review'', Dec. 21, 2009 </ref>

===Language===
The French language derives from the vernacular Latin spoken by the Romans in [[Gaul]], although it includes some Celtic and Germanic words. Historically, French has been used as the international language of diplomacy and commerce. Today it remains one of six official languages at the [[United Nations]] and has been a unifying factor in Africa, Asia, the Pacific, and the Caribbean.
[[Image:Sacre Coeur.jpg|thumb|Le Sacre Coeur.]]
The French language is controlled by the French Academy (''L'Académie française''). As Members of the Academy are appointed for life and this results in a very conservative approach to the changing demands on the language even going as far as banning words with an English origin. The French people (and other French speakers) often ignore the "correct" French and adopt modern English words. This can cause an even wider gap opening up between the cultural elite, who prefer to follow the official French even if it is many years and even decades out of date, and ordinary French speakers who prefer to have a more useful language. There have even been arguments between the government and the Academy over the introduction of "foreign" French words from countries like Belgium and Canada which the government is keen to use. Some even say that this painfully slow and traditional organization is strangling the French language.

===Religion===

The French state is non-religious - it has no sanctioned religion and is forbidden to do so by law, a practice which commonly referred to as ''laicité''. Since the beginning of the 20th century, state banning of religion in education and other state-funded activities created a large proportion of people who are atheists. It was common for teachers to preach atheism at schools and classroom prayer was forbidden.
[[File:Notre-dame.jpg|thumb|left|125px|Notre Dame.]][[File:La Mosquée de Paris.jpg|thumb|left|125px|La Mosquée de Paris.]]
[[File:Muslims in Europe map.jpg|thumb|290px|Muslims in Europe by country.]]
In a January 2007 poll, 51% percent of respondents indicated they were Roman Catholic, including those who never attend religious services. Of Catholics, only 10% went to Church regularly--mainly to Sunday Mass or christenings. There is a cultural particularism in France : a large percentage (30%) of people, calling themselves Catholics, don't believe in God, nor in any kind of spiritual lifeforce. They endorse Catholicism moral teachings, and view Catholic dogma as a tradition. There are an estimated 5 million to 6 million Muslims (8 to 10% of the population), although estimates of how many of these are practicing vary widely. According to a January 17, 2008 survey in the Catholic daily ''La Croix,'' 39% of Muslims surveyed said they observed Islam's five prayers daily, an increase from 31% in 1994. Mosque attendance for Friday prayers rose to 23%, up from 16 percent in 1994, while Ramadan observance reached 70% compared to 60% in 1994. Drinking alcohol, which Islam forbids, also declined to 34% from 39% in 1994.<ref> Data reported by [http://www.state.gov/g/drl/rls/irf/2008/108446.htm U.S. State Department, 2008]</ref>
Protestants make up 3% of the population, Jewish and Buddhist religious groups 1% each, and Sikhs less than 1 percent.

In April 2008 a practicing Buddhist claimed the current estimate of 600,000 Buddhists strongly underrepresents the actual number of adherents; however, he acknowledged the difficulty in distinguishing between practicing Buddhists and the much larger number of persons who characterize themselves as sympathetic to certain Buddhist principles. Scholars distinguish between "sympathizers" (approximately 5 million), "associates" (defined as having a certain degree of proximity to Buddhism and estimated to number between 100,000 and 150,000), and "practitioners" (approximately 12,000). The biggest Buddhist meditation center in the West is in the region of Touraine, and two Tibetan monasteries in the Auvergne region have trained the largest number of Buddhist monks outside Asia, according to 2002 statistical data.

The Jewish community numbers approximately 600,000. According to press reports, at least 60% of Jews are not highly observant, celebrating at most only the High Holy Days. The large majority of observant Jews are Orthodox. There are small Conservative and Reform congregations as well.

Jehovah's Witnesses reported that 250,000 persons attend their services either regularly or periodically.

Orthodox Christians number between 80,000 and 100,000; the vast majority are associated with the Greek or Russian Orthodox Churches.

Other religious groups include Evangelicals, Christian Scientists, and the Church of Jesus Christ of Latter-day Saints (Mormons). Membership in evangelical churches is growing (with as many as 400,000 adherents, according to February 2007 press reports), including African-style "prosperity" churches, especially in the suburbs of Paris, in large part because of increased participation by African and Antillean immigrants. According to the press, there are approximately 31,000 Mormons. The Church of Scientology has an estimated 5,000 to 20,000 members.

==Government==
[[File:Delacroix La liberte guidant le peuple.jpg|thumb|300px|Commemorating the July Revolution of 1830: [[Eugene Delacroix]], ''La liberte guidant le peuple''.]]
The constitution of the Fifth Republic was approved by public referendum on September 28, 1958. It greatly strengthened the powers of the president in relation to those of Parliament. Under this constitution, presidents have been elected directly for a 7-year term since 1958. Beginning in 2002, the presidential term of office was reduced to 5 years. The president names the prime minister, presides over the cabinet, commands the armed forces, and concludes treaties. Traditionally, presidents under the Fifth Republic have tended to leave day-to-day policy-making to the Prime Minister and government; the five-year term of office is expected to make presidents more accountable for the results of domestic policies.

The president can submit questions to a national referendum and can dissolve the National Assembly. In certain emergency situations, with the approval of parliament, the president may assume [[dictator]]ial powers and rule by decree. The main components of France's executive branch are the president, the prime minister and government, and the permanent bureaucracies of the many ministries. Led by a prime minister, who is the head of government, the cabinet is composed of a varying number of ministers, ministers-delegate, and secretaries of state. Parliament meets for one 9-[[month]] session each year. Under special circumstances the president can call an additional session.

Under the Constitution, the legislative branch has few checks on executive power; nevertheless, the National Assembly can still cause a government to fall if an absolute majority of the total Assembly membership votes to censure. The Parliament is [[bicameral]] with a National Assembly and a Senate. The National Assembly is the principal legislative body. Its deputies are directly elected to 5-year terms, and all seats are voted on in each election. Senators are chosen by an electoral college and, under new rules passed in 2003 to shorten the term, serve for six years, with one-half of the Senate being renewed every three years. (As a transitional measure in 2004, 62 Senators were elected to 9-year terms, while 61 were elected to 6-year terms; subsequently, all terms will be six years.) The Senate's legislative powers are limited; the National Assembly has the last word in the event of a disagreement between the two houses. The government has a strong influence in shaping the agenda of Parliament. The government also can declare a bill to be a question of confidence, thereby linking its continued existence to the passage of the legislative text; unless a motion of censure is introduced and voted, the text is considered adopted without a vote.

A distinctive feature of the French judicial system is that the Constitutional Council protects basic rights when they might be potentially violated by new laws and the Council of State protects basic rights when they might be violated by actions of the state. The Constitutional Council examines legislation and decides whether it conforms to the constitution. Unlike the U.S. Supreme Court, it considers only legislation that is referred to it by Parliament, the prime minister, or the president. Moreover, it considers legislation before it is promulgated. The Council of State has a separate function from the Constitutional Council and provides recourse to individual citizens who have claims against the administration. The Ordinary Courts--including specialized bodies such as the police court, the criminal court, the correctional tribunal, the commercial court, and the industrial court--settle disputes that arise between citizens, as well as disputes that arise between citizens and corporations. The Court of Appeals reviews cases judged by the Ordinary Courts.

Traditionally, decision-making in France has been highly centralized, with each of France's departments headed by a prefect appointed by the central government. In 1982, the national government passed legislation to decentralize authority by giving a wide range of administrative and fiscal powers to local elected officials. In March 1986, regional councils were directly elected for the first time, and the process of decentralization continues, albeit at a slow pace.
===Principal Government Officials===
See [[List of French Presidents]].
*President--[[Nicolas Sarkozy]]
*Prime Minister--François Fillon
*Foreign Minister--Alain Juppé
*Ambassador to the United States--François Delattre
*Ambassador to the United Nations--Gérard Araud

===Political Conditions===
[[File:French Market.jpg|thumb|left|300px|Improving French Market.]]
Since his inauguration in May 2007 as France's sixth president under the Fifth Republic, Nicolas Sarkozy focused his first months in office on improving the performance of France's economy through liberalization of labor markets, higher education and taxes. In the April 22, 2007 first round of presidential elections, Sarkozy, the leader of the center-right, union for a popular Movement (UMP) party, placed first; Socialist candidate Ségolène Royal placed second; centrist François Bayrou placed third; and extremist Jean-Marie Le Pen placed fourth out of a field of 12 candidates. Sarkozy prevailed in the May 6, 2007 second round, defeating Royal by a 53.06% to 46.94% margin. Royal's loss marked the third straight defeat for the Socialist candidate in presidential elections.

President Sarkozy assumed office on May 16, 2007, the last day of Jacques Chirac's official term. Sarkozy named François Fillon Prime Minister. Jean-Louis Borloo, the second highest ranking figure in the government, presides over an expanded Ministry of Environment.. Legislative elections held on June 10 and 17, 2007 gave the UMP a large parliamentary majority. In 2007, key ministers included: Jean-Louis Borloo, Ecology and Sustainable Planning; Michèle Alliot-Marie, Interior; Bernard Kouchner, Foreign Affairs; Christine Lagarde, Economy; Brice Hortefeux, Immigration; Rachida Dati, Justice; and Hervé Morin, Defense.

In electing Nicolas Sarkozy, French voters endorsed the wide-ranging program of reforms--including market-oriented social and economic reforms--that were the focal point of Sarkozy's campaign, implicitly giving him the green light to try and implement these reforms quickly, and allowing a way forward for overcoming France's 2005 rejection of the EU constitutional treaty. By embracing a figure long tagged as "pro-American," French voters also expressed their desire to renew trust in the U.S.-France relationship. During the campaign Sarkozy often ended his stump speeches--evoking Martin Luther King--by calling for a "French dream" of social equality, social mobility, and equal opportunity; and his first speech as President-elect assured his "American friends" that they could rely on France's friendship.

Shortly after taking office, President Sarkozy went to work on a series of reforms to address mounting pressure for short- and long-term restructuring, including reduced government spending, flexibility in the implementation of the 35-hour work week, more labor-market flexibility, less taxation, and further privatization and liberalization of the business sector. French and EU analysts stress that longer-term measures must focus on reducing the future burden of ballooning public pension and health care budgets, as well as reducing labor-related taxes. Government action to initiate such reforms may have contributed to the center-right's poor showing in 2004 regional and European Parliamentary elections, and continues to spark periodic strikes and work stoppages throughout France.

It should be noted that France is, in many ways, much more left-wing and socialist than the U.S.A. Even with what the French would call a right-wing president, there are differences fundamental between the right-wing in France and America. France has a very large state and employs a near majority of the workforce directly and indirectly (like state-owned private companies). France also has a very strong welfare state and comprehensive cover for health, unemployment, pensions, etc. This is expensive and so taxes are relatively high. So while President Chirac and his successor President Sarkozy are right-wing, they are not looking to dismantle these policies even if the latter wants to make them more efficient. Compared to America, Sarkozy would be roughly equivalent to a Democrat. The Socialist Party, the opposition political party at the moment, is very strong. There is no comparison to a main American political party.

<blockquote>
During his first year and a half in office, Sarkozy eliminated income taxes on overtime hours, lengthened the contribution period for retirees to receive full pensions, and established a "minimum service" requirement on strike days, among other reforms. He also completed a major revision of the French constitution, which made the president more accountable to Parliament and strengthened the power of the legislature. French and EU analysts stress that longer-term reform measures must focus on reducing the future burden of ballooning public pension and health care budgets, as well as reducing labor-related taxes.
<ref> [http://aadorjan.com/Moin/index.cgi/Political_Structure Political Structure] </ref>
</blockquote>

The coming presidential elections in 2012 highlights the current political situation. It should be noted, that in contrary to the two party system such as in the US, French elections saw a multitude of political movements face each other . The historical right movement (UMP) lead by President Sarkozy will face several opponents. The Parti Socialiste, the main left polical party, whose candidate is François Hollande, as well as the Front National, a far right party, whose popularity is rising due to the economical stagnation, Nicolas Sarkozy's low rate of approval in recent surveys, and the focus of the government on immigration and interior security topics, which have been taken up by the far right.
Several others movements, like the ecologists, the centrists, the communists, the monarchists present candidates, but few are expected to get more than 10% of votes.

===Foreign Relations===
A charter member of the United Nations, France holds one of the permanent seats in the Security Council and is a member of most of its specialized and related agencies. France is also America's oldest ally; French military intervention was instrumental in helping Britain's American colonies establish independence. Because many battles in which the United States was involved during World War I and World War II took place in France, more American soldiers have been killed on French soil than on that of any other foreign country.
[[File:France Mexico relations.jpg|thumb|270px|A court decision has weakened Mexico-France diplomatic relations.]]
France is a leader in Western Europe because of its size, location, and large economy, membership in European organizations, strong military posture, and energetic diplomacy. France generally has worked to strengthen the global economic and political influence of the EU and its role in common European defense. It views Franco-German cooperation and the development of a European Security and Defense Policy (ESDP) with other EU members, as the foundation of efforts to enhance European security.

France supports Quartet (U.S.-EU-Russia-UN) efforts to implement the Middle East roadmap, which envisions establishment of a Palestinian state, living side-by-side in peace and security with Israel. Recognizing the need for a comprehensive peace agreement, France supports the involvement of all Arab parties and Israel in a multilateral peace process.

Since 2003, France has supported four UN Security Council (UNSC) resolutions on Iraq, including UNSCR 1546, which laid out a timetable for Iraq's political transition and reaffirmed UNSC authorization for a Multinational Force in Iraq, at the invitation of the Iraqi government, to stabilize the country. France contributed to the 230 million euro EU contribution to Iraq reconstruction in 2003. After the Iraqi Interim Government took power, France agreed to substantial debt relief and offered police training to Iraqi security forces. In 2006, France and the U.S. collaborated closely to create a consensus in the UN to adopt UNSCR 1696 demanding action from Iran to end its enrichment-related and preprocessing activities.

France continues to play an important role in Africa, especially in its former colonies, through aid programs, commercial activities, military agreements, and cultural impact. In those former colonies where the French presence remains important, France has supported political, military, and social stability. France maintains permanent military bases in Côte d'Ivoire, Djibouti, Gabon, and Senegal, and has maintained a long-term military presence in Chad. An attack on French forces in Côte d'Ivoire in 2004 led to the departure of thousands of French nationals from that country. France responded to the crisis in Côte d'Ivoire by dispatching Operation Unicorn, which has worked with UN forces to help stabilize Côte d'Ivoire. In 2011, during the conflict opposing Gbagbo and Ouattara following the presidental elections, France fully supported Ouattara. France has also deployed forces to Togo (in support of Operation Unicorn in Côte d'Ivoire) and to the Central African Republic, where French forces have assisted government forces in deterring rebel elements. France with EU partners, participated in an international military operation (EUFOR) in the Democratic Republic of the Congo in 2006, which played an important presence during in elections that year in the D.R.C. France was instrumental in organizing the June 2007 ministerial conference on Darfur and has shown leadership in increasing international interest and involvement in Chad and the Central African Republic, both of which are affected by the crisis in Darfur.

France has extensive political and commercial relations with Asian countries, including China, Japan, and Southeast Asia as well as an increasing presence in regional fora. France is seeking to broaden its commercial presence in China and will pose a competitive challenge to U.S. business, particularly in aerospace, high-tech, and luxury markets. In Southeast Asia, France was an architect of the 1991 Paris Accords, which ended the conflict in Cambodia.

===Relations with the United States===

Relations between the United States and France are active and cordial. Mutual visits by high-level officials are conducted frequently. Bilateral contact at the cabinet level has traditionally been active. France and the United States share common values and have parallel policies on most political, economic, and security issues. Differences are discussed frankly and have not generally been allowed to impair the pattern of close cooperation that characterizes relations between the two countries.
[[File:President Obama and President Sarkozy 2009.jpg|thumb|320px|President Obama and President Sarkozy, 2009.]]
France is one of NATO's top three troop contributors. The French support NATO modernization efforts and are leading contributors to the NATO Response Force (NRF). France is keen to build European defense capabilities, including through the development of EU battle-group sized force packages and joint European military production initiatives. Defense Minister Morin supports development of a European defense that must complement, not compete with, NATO, which remains at the core of transatlantic security. French resistance, however, to efforts to reinforce NATO's reach beyond the North Atlantic region remain a source of contention.

France is a close partner with the U.S. in the war on terror. It cooperates with the U.S. to monitor and disrupt terrorist groups and has processed numerous U.S. requests for information under the Mutual Legal Assistance Treaty. French security and intelligence services have rounded up hundreds of extremists in the past year. The French judiciary has upheld the pre-trial detainment of the four French former Guantanamo detainees. France is a strong partner in multiple non-proliferation fora and is a key participant in the Proliferation Security Initiative. Through the "EU3" (France, the U.K., and Germany), France is working to prevent Iran from developing nuclear weapons.

France opposed the use of force in Iraq in March 2003 and did not join the U.S.-led coalition that liberated the country from the dictatorial rule of Saddam Hussein with whom they, like other countries had close economic ties, especially in the supply of nuclear technology.Donald Rumsfeld famously quipped that "going to war without the French is like going on a duck hunt without an accordion".

Americans generally took France's refusal to join in the war very badly. Many politicians, especially Republicans, were incensed and even started campaigns to ridicule France and its military. There was even a campaign to change the name of French fries to Freedom fries. Since no weapons of mass destruction were found after the Iraq war, some have speculated that France was right all along. But many Americans still take a very negative view of the French generally and old stereotypes of rude, unwashed, horse-eating and cowardly French remain a popular vision.

Despite differences over Iraq, the U.S. and France continue to cooperate closely on many issues, most notably the global war on terrorism, efforts to stem the [[proliferation]] of weapons of mass destruction (WMD), and on regional problems, including in Africa, Lebanon, and Kosovo. On Iraq, the French agreed to generous debt relief for Iraq in Paris Club negotiations and have accepted the establishment of a NATO training mission there.

In the Israeli-Palestinian conflict, France seeks robust U.S. engagement in the peace process. France is working to contain the Hamas-led challenge to the Palestinian Authority. President Sarkozy, like his predecessor, President Chirac, is committed to keeping France in supportive relations with Israel.

The U.S. and France have worked closely to support a sovereign and independent Lebanon, free of Syrian domination, having fought with uncharacteristic tenacity to keep the British out in World War II, but without success.
The U.S. and France co-sponsored in September 2004 UNSCR 1559, which called for full withdrawal of Syrian forces, a free and fair electoral process, and disbanding and disarmament of all Lebanese and non-Lebanese militias. In the wake of the assassination of former Lebanese Prime Minister Rafiq Hariri in February 2005, the U.S. and France reiterated calls for a full, immediate withdrawal of all Syrian troops and security services from Lebanon. France also co-sponsored UNSCR 1701 and was one of the leading countries in Europe working to end hostilities between Israel and Hezbollah in 2006 by committing 2,000 troops to UNIFIL-plus. Strong French backing led to adoption of UNSCR 1757 establishing a Special Tribunal for Lebanon to prosecute the perpetrators of the Hariri assassination and other killings of critics of Syria's interference in Lebanon. Foreign Minister Kouchner is working hard to help competing Lebanese political factions agree to a framework for governing the country in accordance with the country's constitution and free from external interference. France also participates in the U.S. Broader Middle East and North Africa initiative.

Trade and investment between the U.S. and France are strong. On average over $1 billion in commercial transactions take place between France the U.S. every day. In 2006 U.S. exports to France totaled over 24.2 billion euros and U.S. imports from France were valued at 37.1 billion euros. France is the eighth partner for total trade of goods (imports plus exports) and the tenth-ranked supplier worldwide to the U.S. and its 10th largest customer. The U.S. is France's third-ranked supplier and its fifth largest customer. There are approximately 2,300 French subsidiaries in the U.S., providing more than 481,100 jobs and generating an estimated $178 billion turnover. U.S. companies in France employ about 603,400 French citizens. The U.S. is the top destination for French investments worldwide.

Relations at a cultural and popular level are complex. While many Americans see the French as too quick surrender in the face of military conflict, many French people view suspiciously America's willingness to go to war. There are suspicions that America will only fight to secure wealth primarily oil or its ideology of the US version of liberty with disregard for other nation's views. Americans often view French culture as being very elitist and overly reliant on the high arts while some French take exception to American "cultural imperialism" even though the French are keen consumers of American-style fast food, cinema and modern music. The French are generally proud of the donation of the Statue of Liberty but can sometimes appear ungrateful for the help America and other Allies gave in World War 2.

==Security Issues==
[[File:French July 14.jpg|right|200px]]
French military doctrine is based on the concepts of national independence, nuclear deterrence, and military sufficiency. France is a founding member of the North Atlantic Treaty Organization ([[NATO]]), and has worked actively with Allies to adapt NATO, internally and externally, to the post-Cold War environment. However, in 1966, the French withdrew from NATO's military bodies while remaining full participants in the alliance's political councils. In December 1995, France announced that it would increase its participation in NATO's military wing, including the Military Committee. France remains a firm supporter of the Organization for Security and Cooperation in Europe (OSCE) and other efforts at cooperation.

Outside of NATO, France has actively and heavily participated in a variety of peacekeeping/coalition efforts in Africa, the Middle East, and the Balkans, often taking the lead in these operations. France has undertaken a major restructuring to develop a professional military that will be smaller, more rapidly deployable and better tailored for operations outside of mainland France. Key elements of the restructuring include reducing personnel, bases, and headquarters and rationalizing equipment and the armament industry. French active-duty military in June 2007 numbered about 350,000 (including Gendarmes), of which nearly 34,000 were deployed outside of French territory. France completed the move to all-professional armed forces when conscription ended on December 31, 2002.
[[File:Nato - airpower.jpg|thumb|left|300px|NATO-airpower.]]
France places a high priority on arms control<ref>[http://archive.newsmax.com/archives/articles/2004/11/10/151219.shtml French Arms Sales to China]</ref> and non-proliferation. After conducting a final series of six nuclear tests, the French signed the Comprehensive Test Ban Treaty in 1996. France has implemented a moratorium on the production, export, and use of anti-personnel landmines and supports negotiations leading toward a universal ban. France is an active participant in the major supplier regimes designed to restrict transfer of technologies that could lead to proliferation of weapons of mass destruction: the Nuclear Suppliers Group, the Australia Group (for chemical and biological weapons), the Non-Proliferation Treaty, and the Missile Technology Control Regime. France participates actively in the Proliferation Security Initiative, and is engaged with the U.S., both bilaterally and at the International Atomic Energy Agency (IAEA) and Organization for the Prohibition of Chemical Weapons (OPCW), to curb nuclear, biological, and chemical (NBC) proliferation from the D.P.R.K., Iran, Libya, and elsewhere. France has joined with the U.S., Germany, and the other three permanent members of the UN Security Council to offer a package of incentives and disincentives to Iran to halt its uranium enrichment activities. France has also signed and ratified the Chemical Weapons Convention. France maintains a color-coded security system, similar to that of the U.S., consisting of yellow, orange, red and scarlet threat levels.

At least 62 French soldiers have been killed in [[Afghanistan]] during [[Operation Enduring Freedom]] since its inception in 2001.<ref>[http://www.defense.gouv.fr/operations/afghanistan/in-memoriam/in-memoriam In Memoriam]</ref>

==Economy==
[[File:Paris-13eme - panorama banlieues.jpg|center|Paris.]]
With a GDP of approximately $2.5 trillion, France is the sixth-largest economy. It has substantial agricultural resources, a large industrial base, and a highly skilled work force. A dynamic services sector accounts for an increasingly large share of economic activity and is responsible for nearly all job creation in recent years. GDP growth was 1.1% in 2003, after two years of steady decline from 3.9% in 2000. GDP growth was 1.7% in 2005, down from 2.5% in 2004 (2000 price basis).

Government economic policy aims to promote investment and domestic growth in a stable fiscal and monetary environment. Creating jobs and reducing the high unemployment rate through recovery-supportive policy has been a top priority. French unemployment dropped from a high of 12% to 8.7% in the late 1990s, and after hovering around 10% during the 2000s, unemployment slipped once again to 8.0% in July 2007. France joined 10 other European Union countries in adopting the euro as its currency in January 1999. Since then, monetary policy has been set by the European Central Bank in Frankfurt. On January 1, 2002, France, along with the other countries of the euro zone, dropped its national currency in favor of euro bills and coins.

Despite significant reform and privatization over the past 15 years, the government continues to control a large share of economic activity: Government spending, at 53.5% of GDP in 2006, is among the highest in the G-7. Regulation of labor and product markets is pervasive. The government continues to own shares in corporations in a range of sectors, including banking, energy production and distribution, automobiles, transportation, and telecommunications.
[[File:Chenonceaux France.jpg|thumb|260px|Chenonceaux.]]
Legislation passed in 1998 shortened the legal work week from 39 to 35 hours for most employees effective January 1, 2000. Recent assessments of the impact of work week reduction on growth and jobs have generally concluded that the goal of job creation was not met. The former administration introduced increasing flexibility into the law, returning the country to a de facto (if not de jure) 39-hour work week in the private sector. Under President Nicolas Sarkozy's impetus, overtime work will be exempt from income taxes on October 1, 2007, a move to encourage work and to increase work duration.

Membership in France's labor unions accounts for approximately 5% of the private sector work force and is concentrated in the manufacturing, transportation, and heavy industry sectors. Most unions are affiliated with one of the competing national federations, the largest and most powerful of which are the communist-dominated General Labor Confederation (CGT), the Workers' Force (FO), and the French Democratic Confederation of Labor (CFDT).

France has been very successful in developing dynamic telecommunications, aerospace, and weapons sectors. With virtually no domestic oil production, France has relied heavily on the development of nuclear power, which now accounts for about 80% of the country's electricity production.

===Trade===
[[Image:French-Perfume.jpg|right|280px]]
France is the second-largest trading nation in Western Europe (after Germany). France ran a $33.7 billion deficit in 2006. Total trade for 2006 amounted to $1,013.5 billion, over 45% of GDP 75.0% of which was with EU-24 countries. In 2003, U.S.-France trade in goods and services totaled $80.3 billion. U.S. industrial chemicals, aircraft and engines, electronic components, telecommunications, computer software, computers and peripherals, analytical and scientific instrumentation, medical instruments and supplies, broadcasting equipment, and programming and franchising are particularly attractive to French importers. Total French trade of goods and services was $1,001 billion in 2006.

Principal French exports to the United States are [[aircraft]] and [[engine]]s, beverages, electrical equipment, chemicals, cosmetics, and luxury products. France is the ninth-largest trading partner of the United States.
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===Agriculture===
[[File:Basque Country France.jpg|left|200px]]
France is the world's second-largest agricultural producer, after the United States. However, the destination of 70% of its exports is other EU member states. Wheat, beef, pork, poultry, and dairy products are the principal exports. The United States, although the second-largest exporter to France, faces stiff competition from domestic production, other EU member states, and third countries. U.S. agricultural exports to France, totaling $464 million in 2003, consist primarily of soybeans and products, feeds and fodders, seafood, and consumer oriented products, especially snack foods and nuts. French agricultural exports to the United States are mainly cheese, processed products, and wine. They amount to about $3.327 billion (2006) annually.

France accounts for about one-third of all agricultural land within the EU. Northern France is characterized by large wheat farms. Dairy products, pork, poultry, and apple production are concentrated in the western region. Beef production is located in central France, while the production of fruits, vegetables, and wine ranges from central to southern France. France is a large producer of many agricultural products and is expanding its forestry and fishery industries. The implementation of the Common Agricultural Policy (CAP) and the Uruguay Round of the GATT Agreement resulted in reforms in the agricultural sector of the economy. Continued revision of the CAP and reforms agreed under the Doha round of World Trade Organization (WTO) will further change French agriculture. France remains Europe's strongest opponent of genetically modified organisms (GMOs) and often assumes an agricultural position at the EU Council to promote this policy.

== See also ==
[[File:Boudin Trouville.jpg|thumb|300px|''Trouville'' by [[Eugene Boudin]], 1864.]]
*[[Gallery of French Masterpieces]]
*[[Famous French paintings]]
*[[French terms]]
*[[Libyan uprising 2011]]

== References ==
{{reflist}}

 
{{Copyright Details (US Government)|http://www.state.gov/r/pa/ei/bgn/3842.htm}}

{{European Union}}

[[Category:European Countries]]
[[Category:NATO members]]
[[Category:France| ]]