A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to certain equations. For example, the equation 2 = − 9 has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem; the idea is to extend the real numbers with an indeterminate i, taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: 2 = 2 = = 9 = − 9, 2 = 2 = 2 = 9 = − 9.
According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers; the 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i; this means that complex numbers can be added and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can be divided by nonzero complex numbers. Overall, the complex number system is a field. Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
The complex number a + bi can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary. A complex number whose imaginary part is zero can be viewed as a real number. Complex numbers can be represented in polar form, which associates each complex number with its distance from the origin and with a particular angle known as the argument of this complex number; the geometric identification of the complex numbers with the complex plane, a Euclidean plane, makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis; the addition of complex numbers is thus depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not available in a vector space. Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1.
For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials; the relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i, which hold for all integers k. The real number a is called the real part of the complex number a + bi. To emphasize, the imaginary part does not include a factor i and b, not bi, is the imaginary part. Formally, the complex numbers are defined as the quotient ring of the polynomia
Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. In particular, 3rd-century astronomers first noted that the ratio of the lengths of two sides of a right-angled triangle depends only of one acute angles of the triangle; these dependencies are now called trigonometric functions. Trigonometry is the foundation of all applied geometry, including geodesy, celestial mechanics, solid mechanics, navigation. Trigonometric functions have been extended as functions of a real or complex variable, which are today pervasive in all mathematics. Sumerian astronomers studied angle measure. They, the Babylonians, studied the ratios of the sides of similar triangles and discovered some properties of these ratios but did not turn that into a systematic method for finding sides and angles of triangles; the ancient Nubians used a similar method.
In the 3rd century BC, Hellenistic mathematicians such as Euclid and Archimedes studied the properties of chords and inscribed angles in circles, they proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. In 140 BC, Hipparchus gave the first tables of chords, analogous to modern tables of sine values, used them to solve problems in trigonometry and spherical trigonometry. In the 2nd century AD, the Greco-Egyptian astronomer Ptolemy constructed detailed trigonometric tables in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today. Centuries passed before more detailed tables were produced, Ptolemy's treatise remained in use for performing trigonometric calculations in astronomy throughout the next 1200 years in the medieval Byzantine and Western European worlds; the modern sine convention is first attested in the Surya Siddhanta, its properties were further documented by the 5th century Indian mathematician and astronomer Aryabhata.
These Greek and Indian works were expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, were applying them to problems in spherical geometry; the Persian polymath Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right. Nasīr al-Dīn al-Tūsī was the first to treat trigonometry as a mathematical discipline independent from astronomy, he developed spherical trigonometry into its present form, he listed the six distinct cases of a right-angled triangle in spherical trigonometry, in his On the Sector Figure, he stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, provided proofs for both these laws. Knowledge of trigonometric functions and methods reached Western Europe via Latin translations of Ptolemy's Greek Almagest as well as the works of Persian and Arab astronomers such as Al Battani and Nasir al-Din al-Tusi.
One of the earliest works on trigonometry by a northern European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus, encouraged to write, provided with a copy of the Almagest, by the Byzantine Greek scholar cardinal Basilios Bessarion with whom he lived for several years. At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond. Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts. Driven by the demands of navigation and the growing need for accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595. Gemma Frisius described for the first time the method of triangulation still used today in surveying, it was Leonhard Euler who incorporated complex numbers into trigonometry.
The works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series. In the 18th century, Brook Taylor defined the general Taylor series. If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees; the two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined; these ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure: Sine function, defined as the ratio of the side opposite the angle to the hypotenuse.
Sin A = opposite hypotenuse = a c. Cosine funct
In probability theory, the normal distribution is a common continuous probability distribution. Normal distributions are important in statistics and are used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be distributed and is called a normal deviate; the normal distribution is useful because of the central limit theorem. In its most general form, under some conditions, it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, they become distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes have distributions that are nearly normal. Moreover, many results and methods can be derived analytically in explicit form when the relevant variables are distributed; the normal distribution is sometimes informally called the bell curve.
However, many other distributions are bell-shaped. The probability density of the normal distribution is f = 1 2 π σ 2 e − 2 2 σ 2 where μ is the mean or expectation of the distribution, σ is the standard deviation, σ 2 is the variance; the simplest case of a normal distribution is known as the standard normal distribution. This is a special case when μ = 0 and σ = 1, it is described by this probability density function: φ = 1 2 π e − 1 2 x 2 The factor 1 / 2 π in this expression ensures that the total area under the curve φ is equal to one; the factor 1 / 2 in the exponent ensures that the distribution has unit variance, therefore unit standard deviation. This function is symmetric around x = 0, where it attains its maximum value 1 / 2 π and has inflection points at x = + 1 and x = − 1. Authors may differ on which normal distribution should be called the "standard" one. Gauss defined the standard normal as having variance σ 2 = 1 / 2, φ = e − x 2 π Stigler goes further, defining the standard normal with variance σ 2 = 1 /: φ = e − π x 2 Every normal distribution is a version of the standard normal distribution whose domain has been stretched by a factor σ and translated by μ: f = 1 σ φ.
The probability density must be scaled by 1 / σ so that the integral is still 1. If Z is a standard normal deviate X = σ Z + μ will have a normal distribution with expected value μ and standard deviation σ. Conversely, if X is a normal deviate with parameters μ and σ 2 Z = / σ
Philosophiæ Naturalis Principia Mathematica
Philosophiæ Naturalis Principia Mathematica referred to as the Principia, is a work in three books by Isaac Newton, in Latin, first published 5 July 1687. After annotating and correcting his personal copy of the first edition, Newton published two further editions, in 1713 and 1726; the Principia states Newton's laws of motion, forming the foundation of classical mechanics. The Principia is considered one of the most important works in the history of science; the French mathematical physicist Alexis Clairaut assessed it in 1747: "The famous book of Mathematical Principles of Natural Philosophy marked the epoch of a great revolution in physics. The method followed by its illustrious author Sir Newton... spread the light of mathematics on a science which up to had remained in the darkness of conjectures and hypotheses."A more recent assessment has been that while acceptance of Newton's theories was not immediate, by the end of a century after publication in 1687, "no one could deny that" "a science had emerged that, at least in certain respects, so far exceeded anything that had gone before that it stood alone as the ultimate exemplar of science generally."In formulating his physical theories, Newton developed and used mathematical methods now included in the field of calculus.
But the language of calculus as we know it was absent from the Principia. In a revised conclusion to the Principia, Newton used his expression that became famous, Hypotheses non fingo. In the preface of the Principia, Newton wrote:... Rational Mechanics will be the sciences of motion resulting from any forces whatsoever, of the forces required to produce any motion proposed and demonstrated... And therefore we offer this work as mathematical principles of his philosophy. For all the difficulty of philosophy seems to consist in this—from the phenomenas of motions to investigate the forces of Nature, from these forces to demonstrate the other phenomena... The Principia deals with massive bodies in motion under a variety of conditions and hypothetical laws of force in both non-resisting and resisting media, thus offering criteria to decide, by observations, which laws of force are operating in phenomena that may be observed, it attempts to cover hypothetical or possible motions both of celestial bodies and of terrestrial projectiles.
It explores. Its third and final book deals with the interpretation of observations about the movements of planets and their satellites, it shows:. The opening sections of the Principia contain, in revised and extended form, nearly all of the content of Newton's 1684 tract De motu corporum in gyrum; the Principia begin with "Definitions" and "Axioms or Laws of Motion", continues in three books: Book 1, subtitled De motu corporum concerns motion in the absence of any resisting medium. It opens with a mathematical exposition of "the method of first and last ratios", a geometrical form of infinitesimal calculus; the second section establishes relationships between centripetal forces and the law of areas now known as Kepler's second law, relates circular velocity and radius of path-curvature to radial force, relationships between centripetal forces varying as the inverse-square of the distance to the center and orbits of conic-section form. Propositions 11–31 establish properties of motion in paths of eccentric conic-section form including ellipses, their relation with inverse-square central forces directed to a focus, include Newton's theorem about ovals.
Propositions 43–45 are demonstration that in an eccentric orbit under centripetal force where the apse may move, a steady non-moving orientation of the line of apses is an indicator of an inverse-square law of force. Book 1 contains some proofs with little connection to real-world dynamics, but there are sections with far-reaching application to the solar system and universe: Propositions 57–69 deal with the "motion of bodies drawn to one another by centripetal forces". This section is of primary interest for its application to the Solar System, includes Proposition 66 along with its 22 corollaries: here Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions, a problem which gained name and fame as the three-body problem. Prop
Central limit theorem
In probability theory, the central limit theorem establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution if the original variables themselves are not distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. For example, suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, that the arithmetic mean of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the distribution of the average will be approximated by a normal distribution. A simple example of this is that if one flips a coin many times the probability of getting a given number of heads in a series of flips will approach a normal curve, with mean equal to half the total number of flips in each series.
The central limit theorem has a number of variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution occurs for non-identical distributions or for non-independent observations, given that they comply with certain conditions; the earliest version of this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is now known as the de Moivre–Laplace theorem. In more general usage, a central limit theorem is any of a set of weak-convergence theorems in probability theory, they all express the fact that a sum of many independent and identically distributed random variables, or alternatively, random variables with specific types of dependence, will tend to be distributed according to one of a small set of attractor distributions. When the variance of the i.i.d. Variables is finite, the attractor distribution is the normal distribution. In contrast, the sum of a number of i.i.d.
Random variables with power law tail distributions decreasing as |x|−α − 1 where 0 < α < 2 will tend to an alpha-stable distribution with stability parameter of α as the number of variables grows. Let be a random sample of size n—that is, a sequence of independent and identically distributed random variables drawn from a distribution of expected value given by µ and finite variance given by σ2. Suppose we are interested in the sample average S n:= X 1 + ⋯ + X n n of these random variables. By the law of large numbers, the sample averages converge in probability and surely to the expected value µ as n → ∞; the classical central limit theorem describes the size and the distributional form of the stochastic fluctuations around the deterministic number µ during this convergence. More it states that as n gets larger, the distribution of the difference between the sample average Sn and its limit µ, when multiplied by the factor √n, approximates the normal distribution with mean 0 and variance σ2.
For large enough n, the distribution of Sn is close to the normal distribution with mean µ and variance σ2/n. The usefulness of the theorem is that the distribution of √n approaches normality regardless of the shape of the distribution of the individual Xi. Formally, the theorem can be stated as follows: Lindeberg–Lévy CLT. Suppose is a sequence of i.i.d. Random variables with E = µ and Var = σ2 < ∞. As n approaches infinity, the random variables √n converge in distribution to a normal N: n → d N. In the case σ > 0, convergence in distribution means that the cumulative distribution functions of √n converge pointwise to the cdf of the N distribution: for every real number z, lim n → ∞ Pr = Φ, where Φ is the standard normal cdf evaluated at x. Note that the convergence is uniform in z in the sense that lim n → ∞ sup z ∈ R | Pr − Φ | = 0, where sup denotes the least upper bound of the se
Saumur is a commune in the Maine-et-Loire department in western France. The historic town is located between the Loire and Thouet rivers, is surrounded by the vineyards of Saumur itself, Bourgueil, Coteaux du Layon, etc. which produce some of France's finest wines. Early settlement of the region goes back many thousands of years; the Dolmen de Bagneux on the south of the town, is 23 meters long and is built from 15 large slabs of the local stone, weighing over 500 tons. It is the largest in France; the Château de Saumur was constructed in the 10th century to protect the Loire river crossing from Norman attacks after the settlement of Saumur was sacked in 845. The castle, destroyed in 1067 and inherited by the House of Plantagenet, was rebuilt by Henry II of England in the 12th century, it changed hands several times between Anjou and France until 1589. Houses in Saumur are constructed exclusively of the Tuffeau stone; the caves dug to excavate the stone have become tunnels and have been used by the local vineyards as locations to store their wines.
Amyraldism, or the School of Saumur, is the name used to denote a distinctive form of Reformed theology taught by Moses Amyraut at the University of Saumur in the 17th century. Saumur is the scene for Balzac's novel Eugénie Grandet, written by the French author in 1833. Prior to the French Revolution Saumur was the capital of the Sénéchaussée de Saumur, a bailiwick, which existed until 1793. Saumur was the location of the Battle of Saumur during the Revolt in the Vendée, becoming a state prison under Napoleon Bonaparte; the town was an equestrian centre with both the military cavalry school from 1783 and the Cadre Noir based there. During the Battle of France, in World War II, Saumur was the site of the Battle of Saumur where the town and south bank of the Loire was defended by the teenage cadets of the cavalry school, to their great credit and for the Honour of France. In 1944 it was the target of Azon bombing raids by Allied planes; the first raid, on 8/9 June 1944, was against a railway tunnel near Saumur, seeing the first use of the 12,000 lb Tallboy "earthquake" bombs.
The hastily organized night raid was to stop a planned German Panzer Division, travelling to engage the newly landed allied forces in Normandy. The panzers were expected to use the railway to cross the Loire. No. 83 Squadron RAF illuminated the area with flares by four Avro Lancasters and marked the target at low level by three de Havilland Mosquitos. 25 Lancasters of No. 617 Squadron RAF, the "Dambusters" dropped their Tallboys from 18,000 ft with great accuracy. They hit the approaches to the bridge, blocked the railway cutting and one pierced the roof of the tunnel, bringing down a huge quantity of rock and soil which blocked the tunnel, badly delaying the German reinforcements moving towards Normandy 2nd SS Panzer Division Das Reich; the damaged tunnel was dug out to make a deeper cutting, resulting in the need for a second attack. On 22 June, nine Consolidated B-24 Liberators of the United States Army Air Forces used the new Azon 1,000 lb glide bombs against the Saumur rail bridge, they failed to destroy the bridge.
During the morning of 24 June, 38 American Boeing B-17 Flying Fortresses with conventional bombs attacked the bridge. The bridge was damaged; the town of Saumur was awarded the Croix de Guerre with palm for its resistance and display of French patriotism during the war. Saumur is home to the Cadre Noir, the École Nationale d'Équitation, known for its annual horse shows, as well as the Armoured Branch and Cavalry Training School, the officer school for armored forces. There is the national tank museum, the Musée des Blindés, with more than 850 armored vehicles, wheeled or tracked. Most of them are from France, though some come from other countries such as Brazil and the Soviet Union, as well as axis and allied vehicles of World War Two; the annual military Carrousel takes place in July each year, as it has done for over 160 years, with displays of horse cavalry skills and modern military vehicles. Amongst the most important monuments of Saumur are the great Château de Saumur itself which stands high above the town, the nearby Château de Beaulieu which stands just 200 metres from the south bank of the Loire river and, designed by the architect Jean Drapeau.
A giant sequoia tree stands in the grounds of Château de Beaulieu. The Dolmen de Bagneux is on the old road going south; the architectural character of the town owes much to the fact that it is constructed exclusively of the beautiful, but fragile, Tuffeau stone. The wine industry surrounds Saumur, many utilising the tunnels as cellars with the hundreds of domaines producing white, rosé and sparkling wines. Visits to producers and the annual Grandes Tablées du Saumur-Champigny is a popular annual event held in early August with over 1 km of tables set up in Saumur so people can sample the local foods and wine. Saumur has a famous weekly market; every Saturday morning with hundreds of stalls open for business in the streets and squares of the old town, from before 8am. Its skyline has been compared with that of the capital of Slovakia. Saumur was the birthplace of: Anne Le Fèvre Dacier and translator of classics Jeanne Delanoue, made a Roman Catholic Saint in 1982 François Bontemps, General of the French Revolutionary Wars.
Charles Ernest Beulé, archeologist Coco Chanel, fashion designer Yves Robert, composer, writer, producer Jack le Goff, equestrian Fanny Ardant, actr
Kingdom of France
The Kingdom of France was a medieval and early modern monarchy in Western Europe. It was one of the most powerful states in Europe and a great power since the Late Middle Ages and the Hundred Years' War, it was an early colonial power, with possessions around the world. France originated as West Francia, the western half of the Carolingian Empire, with the Treaty of Verdun. A branch of the Carolingian dynasty continued to rule until 987, when Hugh Capet was elected king and founded the Capetian dynasty; the territory remained known as Francia and its ruler as rex Francorum well into the High Middle Ages. The first king calling himself Roi de France was Philip II, in 1190. France continued to be ruled by the Capetians and their cadet lines—the Valois and Bourbon—until the monarchy was overthrown in 1792 during the French Revolution. France in the Middle Ages was a feudal monarchy. In Brittany and Catalonia the authority of the French king was felt. Lorraine and Provence were states of the Holy Roman Empire and not yet a part of France.
West Frankish kings were elected by the secular and ecclesiastic magnates, but the regular coronation of the eldest son of the reigning king during his father's lifetime established the principle of male primogeniture, which became codified in the Salic law. During the Late Middle Ages, the Kings of England laid claim to the French throne, resulting in a series of conflicts known as the Hundred Years' War. Subsequently, France sought to extend its influence into Italy, but was defeated by Spain in the ensuing Italian Wars. France in the early modern era was centralised. Religiously France became divided between the Catholic majority and a Protestant minority, the Huguenots, which led to a series of civil wars, the Wars of Religion. France laid claim to large stretches of North America, known collectively as New France. Wars with Great Britain led to the loss of much of this territory by 1763. French intervention in the American Revolutionary War helped secure the independence of the new United States of America but was costly and achieved little for France.
The Kingdom of France adopted a written constitution in 1791, but the Kingdom was abolished a year and replaced with the First French Republic. The monarchy was restored by the other great powers in 1814 and lasted until the French Revolution of 1848. During the years of the elderly Charlemagne's rule, the Vikings made advances along the northern and western perimeters of the Kingdom of the Franks. After Charlemagne's death in 814 his heirs were incapable of maintaining political unity and the empire began to crumble; the Treaty of Verdun of 843 divided the Carolingian Empire into three parts, with Charles the Bald ruling over West Francia, the nucleus of what would develop into the kingdom of France. Charles the Bald was crowned King of Lotharingia after the death of Lothair II in 869, but in the Treaty of Meerssen was forced to cede much of Lotharingia to his brothers, retaining the Rhone and Meuse basins but leaving the Rhineland with Aachen and Trier in East Francia. Viking advances were allowed to increase, their dreaded longships were sailing up the Loire and Seine rivers and other inland waterways, wreaking havoc and spreading terror.
During the reign of Charles the Simple, Normans under Rollo from Norway, were settled in an area on either side of the River Seine, downstream from Paris, to become Normandy. The Carolingians were to share the fate of their predecessors: after an intermittent power struggle between the two dynasties, the accession in 987 of Hugh Capet, Duke of France and Count of Paris, established the Capetian dynasty on the throne. With its offshoots, the houses of Valois and Bourbon, it was to rule France for more than 800 years; the old order left the new dynasty in immediate control of little beyond the middle Seine and adjacent territories, while powerful territorial lords such as the 10th- and 11th-century counts of Blois accumulated large domains of their own through marriage and through private arrangements with lesser nobles for protection and support. The area around the lower Seine became a source of particular concern when Duke William took possession of the kingdom of England by the Norman Conquest of 1066, making himself and his heirs the King's equal outside France.
Henry II inherited the Duchy of Normandy and the County of Anjou, married France's newly divorced ex-queen, Eleanor of Aquitaine, who ruled much of southwest France, in 1152. After defeating a revolt led by Eleanor and three of their four sons, Henry had Eleanor imprisoned, made the Duke of Brittany his vassal, in effect ruled the western half of France as a greater power than the French throne. However, disputes among Henry's descendants over the division of his French territories, coupled with John of England's lengthy quarrel with Philip II, allowed Philip II to recover influence over most of this territory. After the French victory at the Battle of Bouvines in 1214, the English monarchs maintained power only in southwestern Duchy of Guyenne; the death of Charles IV of France in 1328 without male heirs ended the main Capetian line. Under Salic law the crown could not pass through a woman (Philip IV's daughter