archive-eu.com » EU » M » MATHEMATICS-IN-EUROPE.EU Total: 1028 Choose link from "Titles, links and description words view": Or switch to
"Titles and links view". |

- Mathematics In Europe - Anasayfa

right cdots quad 1 frac12 frac13 frac14 cdots end align Now Leibniz subtracts from both sides the sum 1 1 2 1 3 cdots But that is forbidden since the sum is infinite and the difference of infinite quantities is indeterminate In this case the result happens to be correct It was only a century later that mathematicians such as Augustin Cauchy put the foundations of analysis on a firm footing Translated by David Kramer Pierre Simon Laplace March 3 1749 March 5 1827 by Heinz Klaus Strick Germany Ayrıntılar Kategori Strick France 1955 Born the son of a farmer and cider producer in Beaumont en Auge Normandy he attended a local Benedictine school and then the Jesuit college in Caen with the goal typical for a child of the Third Estate of a career in the service of the Church However his teachers noted his unusual mathematical ability and in 1768 without having finished his schooling Laplace went off to Paris with a letter of introduction to Jean Baptiste le Rond d Alembert 1717 1783 the most influential member of the Académie des Sciences At first d Alembert took no notice of the letter but Laplace won him over with his confident and competent demeanor D Alembert supported and encouraged the talented student who soon gained notice with publications on extreme value problems differential equations probability theory and astronomy At the age of 22 Laplace made his first application for membership in the Académie When the application was denied d Alembert obtained for him a position as a mathematics teacher at the Paris Military Academy When his application to the Académie again failed the following year Laplace asked Joseph Louis Lagrange 1736 1813 Euler s successor as director of the mathematics division of the Prussian Academy of Sciences whether there might be a position for him at Berlin But before he received a reply Laplace was offered a permanent position as alternate member of the Paris Academy In the years preceding the French Revolution Laplace developed into one of the most influential French scientists His mentor d Alembert began more and more to sense that his own life s work was diminishing in significance Laplace did not hide his light under a bushel He dominated discussions in the Academy taking a firm line on all subjects even the nonmathematical In 1784 he was named examiner at the military academy He put his position to good use making contacts in administrative circles the officer class generally came from the nobility Among the students whom he examined was the sixteen year old Napoleon Bonaparte Jean Baptiste le Rond d Alembert France 1959 In 1790 he became a member of a commission with the charge of standardizing the units of measurement and at the same time to introduce units based on the decimal system In 1793 he fled Paris with his wife who was 20 years his junior and their two children thereby avoiding the fate of his colleague Antoine Laurent de Lavoisier 1743 1794 who was beheaded After the end of the Jacobin Terror he returned and took over the leadership of the Bureau of Longitude and the Paris Observatory The Académie des Sciences was reopened as the Institut National des Sciences et des Arts Antoine Laurent de Lavoisier France 1943 Together with the chemist Claude Louis Berthollet 1748 1822 he began in 1806 to assemble in his hometown of Arcueil near Paris together with the Société d Arcueil a circle of researchers to which belonged Alexander von Humboldt During Napoleon s reign Laplace and Berthollet controlled the academic life in France thanks to their closeness to the emperor Claude Louis Berthollet France 1958 Laplace focused much of his research effort on investigating the application of mathematics to questions in astronomy Observations of the orbits of Jupiter and Saturn indicated a destabilizing influence in the solar system Isaac Newton had hypothesized that the solar system can maintain its equilibrium through regular divine intervention In 1776 Laplace discovered that the equations of motion established by Euler and Lagrange for the motions of the planets were not sufficiently precise and that the terms omitted in their calculations could have a significant effect over the course of a number of years He proved that from a long term viewpoint the solar system is indeed in equilibrium In 1796 there appeared a general work the Exposition du système du monde which contains the following hypothesis on the origin of the solar system Gas clouds broke away from the Sun and condensed to form the planets Similar ideas had been formulated by Immanuel Kant 1724 1804 in the year 1755 Alexander von Humboldt Federal Republic of Germany 1959 After an intensive involvement with a variety of problems in astronomy Laplace finally collected the results of his research in a comprehensive book Beginning in 1799 he published the five volumes of Mécanique céleste in which as the physicist Jean Baptiste Biot 1744 1862 noted Newton s Principia was translated into the language of calculus When Napoleon observed that God is nowhere to be found in the Mécanique céleste Laplace is said to have answered Sire je n avais pas besoin de cette hypothèse Your Majesty I had no need of such a hypothesis Laplace has also to his credit important advances in the theory of probability In his Théorie analytique des probabilités 1812 and in the more generalist Essai philosophique sur les probabilités 1814 he adopts the definition today considered classical of probability given by Abraham de Moivre 1667 1754 a Frenchman who had emigrated to England Doctrine of Chances 1738 probability of an event number of favorable cases number of possible cases Mozambique 2001 This approach of equiprobability seemed to him justified insofar as nothing compels us to believe that one of the cases must occur more easily than the others This textbook concerned itself intensively with the investigation of games of chance but also with conditional and independent events here he acknowledged the contributions of Thomas Bayes 1702 1761 on conditional probabilities with actuarial statistics with geometric probabilities with Bernoulli s law of large numbers with special cases of the central limit theorem de Moivre and Laplace theorems as well as the method of least squares For Laplace there exist questions of probability only because we lack complete knowledge He was convinced that if some intelligence were able to know at a certain moment all the forces and motions of all particles and could use all the methods of mathematical analysis then all motions could be accurately predicted Nothing would be uncertain for such an intelligence and both the future and the past would lie clearly before its eyes This hypothetical intelligence which embodies the fundamental idea of causal determinism is known in philosophy as Laplace s demon Laplace doubtlessly belongs among the most important mathematicians of all time in particular his contributions to probability theory and celestial mechanics were taught unchanged for decades One should mention as well his articles on determinants Laplace expansion differential equations Laplace operator and theoretical physics Laplace equation Part of his genius lay in the capacity to recognize useful approaches for further development of an idea in the work of others His use of such sources however was somewhat problematic in that his frequently adopted an idea without acknowledging its source Throughout his life Laplace changed his political affiliations frequently On account of this opportunistic behavior he increasingly lost respect After a period of active support by the republican regime he became a supporter of Consul Napoleon who appointed him for six weeks to the post of interior minister then to membership in the senate and later to the vice presidency which gave him a substantial income As emperor Napoleon bestowed on Laplace the title of count As Napoleon s star began to sink Laplace shifted his loyalty to the Bourbons who rewarded him with the title of marquis and peer of Paris In contrast to other famous French intellectuals such as Lagrange for example Laplace was not laid to rest in the Panthéon in Paris Translated by David Kramer Carl Friedrich Gauss April 30 1777 February 23 1855 by Heinz Klaus Strick Germany Ayrıntılar Kategori Strick Federal Republic of Germany 1955 Even during his lifetime the Braunschweig Brunswick native Carl Friedrich Gauss was called princeps mathematicorum the prince of mathematics The number of his important mathematical discoveries is truly astounding His unusual talent was recognized when he was still in elementary school It is told that the nine year old Gauss completed almost instantly what should have been a lengthy computational exercise The teacher one Herr Büttner had presented to the class the addition exercise 1 2 3 cdots 100 Gauss s trick in arriving at the sum 5050 was this Working from outside to inside he calculated the sums of the biggest and smallest numbers 1 100 2 99 3 98 50 51 which gives fifty times 101 Herr Büttner realized that there was not much he could offer the boy and so he gave him a textbook on arithmetic which Gauss worked through on his own Together with his assistant Martin Bartels 1769 1836 Büttner convinced the boy s parents for whom such abilities were outside their ken the father worked as a bricklayer and butcher the mother was practically illiterate that their son absolutely had to be placed in a more advanced school From age 11 Gauss attended the Catherineum high school and at 14 he was presented to Duke Carl Wilhelm Ferdinand von Braunschweig who granted him a stipend that made it possible for him to take up studies at the Collegium Carolinum today the University of Braunschweig So beginning in 1795 Gauss studied mathematics physics and classical philology at the University of Göttingen which boasted a more extensive library His physics professor Georg Christoph Lichtenberg 1742 1799 awakened in Gauss a lifelong interest in experimentation After a period of uncertainty as to what subject he should concentrate in indeed Gauss was also quite talented linguistically a brilliant idea for the solution of a millennia old geometric problem finally on March 30 1796 tipped the balance in favor of mathematics Georg Christoph Lichtenberg Germany 1992 Gauss s insight was the realization of just which regular polygons could be constructed with straightedge and compass a regular n gon is constructible with straightedge and compass if and only if n has as its divisors only powers of 2 and Fermat primes that is primes of the form p 2 2 m 1 quad m in mathbb N 0 Since Gauss s discovery no further progress on this problem has been made namely only five Fermat primes are known and it is unknown whether the number of such primes is finite In particular Gauss was able to give a construction for p 2 2 2 1 17 that is for the regular 17 gon Gauss and the regular 17 gon DDR 1977 It is with this result that Gauss began a diary written in Latin in which by 1814 he had written down 146 discoveries This diary was discovered only years after his death and published much later Many priority disputes were thereby resolved in Gauss s favour For example one knows from the diary entries that even in his adolescence Gauss had worked on problems of the distribution of prime numbers Perhaps the continual use of tables of logarithms as a calculational aid and particularly calculation with the number ln 10 2 3 led Gauss at age 15 to the conjecture that pi x the number of primes less than x is approximately proportional to x ln x a statement whose precise proof was achieved only one hundred years later x 10 100 1000 10 000 100 000 1 000 000 10 000 000 pi x 4 25 168 1229 9592 78 498 664 579 frac x pi x 2 5 4 0 6 0 8 1 10 4 12 7 15 0 increase 2 0 2 1 2 3 2 3 2 3 One also finds the word epsilon upsilon rho eta kappa alpha eureka I have found it in a diary entry from 1796 together with text num Delta Delta Delta Gauss had discovered a proof of a conjecture made by Pierre de Fermat every positive integer can be represented as the sum of at most three triangular numbers where the sequence of triangular numbers is given by 1 3 6 10 15 dotsc Visualization of the triangular numbers At the University of Göttingen Gauss became friends with Farkas Bolyai 1775 1856 with whom he maintained contact throughout his life Farkas Bolyai Hungary 1975 He ended his studies without taking any examinations however his sponsor and sovereign insisted that he take his doctoral degree from the domestic University of Helmstedt His dissertation written in Latin of 1799 is dedicated with enormous gratitude to his sovereign Serenissimo Principi ac Domino Carolo Guilielmo Ferdinando His dissertation adviser was the most respected German mathematician of the time Johann Friedrich Pfaff 1765 1825 In his dissertation Gauss takes a critical look at the proof of the fundamental theorem of algebra published by Jean le Rond d Alembert 1717 1783 and then gives a rigorous proof avoiding however the use of complex numbers which in his opinion still had no right of citizenship in mathematics In the course of his life he produced an additional three proofs of this theorem but now using calculation with complex numbers The fundamental theorem of algebra states that every polynomial equation of degree n has precisely n solutions in the set of complex numbers The Gaussian numbers Federal Republic of Germany 1977 On receiving his doctorate Gauss was granted an allowance by the duke while he worked on his book Disquisitiones arithmeticae Arithmetic Investigations As an aside in 1800 he published an algorithm for computing the date on which Easter falls From the year x one first calculates several auxiliary quantities a equiv x bmod 19 b equiv x bmod 4 c equiv x bmod 7 k equiv x operatorname div 100 p equiv 8k 13 operatorname div 25 q equiv k operatorname div 4 M equiv 15 k p q bmod 30 N equiv 4 k q bmod 7 d equiv 19a M bmod 30 e equiv 2b 4c 6d N bmod 7 Then Easter falls on the 22 d e th day of March If this number is greater than 31 then one must subtract 31 to obtain the date of Easter in April The formula has the following exceptions If d e 35 then Easter falls on April 19 if d 28 e 6 and a 10 then Easter falls on April 18 We have reproduced here the version of the algorithm from 1816 in the intervening years Gauss had introduced computation with congruences in particular the modulo notation a equiv b bmod n read a is congruent to b modulo n means that a and b have the same remainder on division by n a operatorname div b means the quotient a div b with the remainder being ignored The Disquisitiones appeared after a long delay in 1801 Each of the seven chapters taken for itself alone aroused international interest in mathematical circles In his foreword Gauss expressly acknowledged his precursors Pierre de Fermat Leonhard Euler Joseph Louis Lagrange 1736 1813 and above all Adrien Marie Legendre 1752 1833 whose book Essai sur la théorie des nombres of 1797 had an unfortunate temporal overlap with the Disquisitiones The individual chapters deal with the theory of arithmetic and congruences with the proof of Leonhard Euler s 1707 1783 conjecture on quadratic reciprocity with the theory of quadratic forms the solution of equations of the form ax 2 2bxy cy 2 m with continued fractions and primality tests and with the solution of equations of the form x n 1 n in mathbb N and x m equiv 1 bmod p x 0 1 x 1 frac 12 i cdot frac sqrt 3 2 x 2 frac 12 i cdot frac sqrt 3 2 Solutions of x 3 1 x 0 1 x 1 i x 2 1 x 3 i Solutions of x 4 1 x 0 1 x 1 frac sqrt 5 1 4 i cdot frac sqrt 2 sqrt 5 10 4 x 2 frac sqrt 5 1 4 i cdot frac sqrt 2 sqrt 5 10 4 x 3 frac sqrt 5 1 4 i cdot frac sqrt 2 sqrt 5 10 4 x 4 frac sqrt 5 1 4 i cdot frac sqrt 2 sqrt 5 10 4 Solutions of x 5 1 x 0 1 x 1 frac 12 i cdot frac sqrt 3 2 x 2 frac 12 i cdot frac sqrt 3 2 x 3 1 x 4 frac 12 i cdot frac sqrt 3 2 x 5 frac 12 i cdot frac sqrt 3 2 Solutions of x 6 1 The law of quadratic reciprocity describes the conditions under which quadratic congruence equations are solvable If p and q are prime numbers then the two congruence equations x 2 equiv p bmod q and x 2 equiv q bmod p are either both solvable or both not solvable unless p and q each have remainder 3 on division by 4 in which case one equation is solvable while the other is not Here are a few examples The equation x 2 equiv5 bmod7 is not solvable that is there are no perfect squares in the sequence 5 12 19 26 33 40 dotsc since the reciprocal equation x 2 equiv 7 bmod 5 equiv 2 bmod 5 has no solution There is no square whose final digit is 2 or 7 The equation x 2 equiv5 bmod11 has a solution since it is clear that perfect squares appear in the sequence of numbers 5 16 27 38 49 60 dotsc But then the reciprocal equation x 2 equiv11 bmod 5 equiv1 bmod 5 must have at least one solution there exist squares with 1 or 6 as final digit The equation x 2 equiv3 bmod 11 is solvable since in the sequence 3 14 25 36 47 58 dotsc there are some perfect squares But then the reciprocal equation x 2 equiv11 bmod 3 equiv2 bmod 3 has no solution that is in the sequence 2 5 8 11 14 17 20 23 dotsc are to be found no perfect squares An equation of the form x n 1 is called a cyclotomic from the Greek kyklos circle temnein to cut equation since one can write down the n solutions in the form x k cos left frac k cdot2 pi n right i cdot sin left frac k cdot2 pi n right with k 0 1 2 dotsc n 1 If one then draws these points as was Gauss s practice after 1820 in the complex plane also known as the Gaussian plane in which the number 1 is drawn to the right of the origin and the number i above it then these points form the vertices of a regular n gon on the unit circle see the figures above Gauss had suddenly become famous As an expression of gratitude to the duke he turned down an invitation to take up residence in Saint Petersburg hoping that his sovereign would build him an observatory in Braunschweig His particular interest in astronomy was strengthened by yet another sensational accomplishment one that made his name known among nonmathematicians as well On January 1 1801 the Italian astronomer Giuseppe Piazzi discovered the asteroid Ceres but then after a couple of days lost all trace of it as the asteroid vanished behind the sun Using Piazzi s data from his sightings Gauss calculated the asteroid s orbit using the method of least squares making possible the rediscovery of Ceres the following year by Heinrich Olbers Gauss had come up with the method of minimizing errors by considering the squares of deviations from a given model when he was only 17 years old He presented the theory behind the method together with a description of its practical application to astronomy in 1809 in the paper Motus corporum coelestium in sectionibus conicis solem ambientium the motion of the celestial bodies that orbit the sun in conic sections Again here there is a priority dispute since Legendre proposed the same method independently of Gauss in 1806 Gauss and Bessel Nicaragua 1994 After the duke of Braunschweig was mortally wounded in 1807 in the Battle of Jena and Auerstedt Gauss had to move to Göttingen to assume a professorship of astronomy in order to provide for himself and his new family He was not pleased that the post included teaching duties and he made every effort to keep them to a minimum His happy marriage to Johanna Osthoff lasted only four years She died giving birth to their third child To provide care for his children he married a friend of Johanna s Minna Waldeck with whom he had an additional three children When his second wife died in 1831 his youngest daughter Therese took care of the household in which Gauss s mother had resided since the death of her husband she died in 1839 at the age of 95 Gaussian bell curve sextant As director of the observatory in Göttingen Gauss worked on improving the design of telescopes and investigated the question of constructing optics with minimal distortion In this capacity he was in frequent contact with the director of the Königsberg observatory Friedrich Wilhelm Bessel 1784 1846 In connection with his astronomical investigations Gauss discovered that random observational errors are normally distributed The graph of the associated function varphi x frac1 sqrt 2 pi cdot e frac x 2 2 the Gaussian distribution has the shape of a bell curve pictured on Germany s last ten mark note valid from 1989 to 2001 In 1797 Gauss had his first experience as a surveyor mapping the Kingdom of Westphalia as ordered by its ruler King Jérôme a brother of Napoleon In 1818 not only did he undertake the direction of a survey of the Kingdom of Hanover but he carried out the actual work over a period of 14 years without considering the associated physical exertion In the process he improved the methods of geodetic surveying in many respects He invented the heliotrope a device that uses mirrors to direct sunlight in the direction of the observer who can be up to one hundred kilometres away He developed the mathematics of error calculation further devised a method for the systematic solution of linear equations Gaussian elimination and invented methods for a minimally distorted representation of the curved surface of the earth using cartographic projection Gaussian coordinates As an example of the use of Gaussian elimination the following system of three equations in three unknowns can be solved by bringing it step by step into triangular form and then solving the equations from bottom to top begin align 1x 2y 1z 8 1x 1y 3z 8 1x 2y 1z 6 end align begin align 1x 2y 1z 8 3y 2z 0 1y 4z 14 end align begin align 1x 2y 1z 8 3y 2z 0 14z 42 end align begin align x 1 y 2 z 3 end align Spurred on by his practical geodetic work Gauss worked intensively on questions of geodetic surveying and differential geometry that is the curvature of surfaces in three dimensional space one measure of curvature is called Gaussian curvature as well as the determination of the volume of curved surfaces which can be computed with the aid of the Gauss integral theorem Already during his geometry classes in school Gauss began to doubt whether Euclidean geometry was the only possible geometric system Do there exist for example geometries in which the angle sum of a triangle is not always equal to 180 Euclid developed his geometry on the basis of systematic deduction from a set of five axioms or postulates The fifth known as the parallel postulate in the plane for every straight line and point external to the line there exists precisely one line passing through the given point and parallel to the given line appears to be less elementary than the other four and for that reason geometers searched over and over and always in vain to derive it from the others Around 1817 Gauss realized that such a derivation was impossible and instead investigated the question what type of geometry one might be able to construct if one threw out the fifth postulate He discussed his methods with his friend Farkas Bolyai but he decided against publishing his ideas After all it was only a few years previously that the philosopher Immanuel Kant had authoritatively declared in his Critique of Pure Reason that the geometry of Euclid was logically necessary and hence incontrovertible Independently of Gauss and of each other the Russian mathematician Nikolai Ivanovich Lobachevsky 1797 1856 and János Bolyai 1802 1860 the son of Farkas Bolyai developed a new geometry without the parallel postulate and have since then been considered the discoverers of non Euclidean geometry the term is due to Gauss János Bolyai Hungary 1960 János was the first to publish in 1823 Lobachevsky who had studied mathematics in Russian Kazan under Martin Bartels the one time assistant in the Braunschweig elementary school gave a lecture in 1826 on an imaginary geometry and wrote a number of articles on the subject These became known in Europe only after 1837 when a paper appeared in French translation Gauss is reported to have said regarding Bolyai s discovery To praise him would simply be to praise myself and on Lobachevsky s publication that he had obtained the same insights 54 years earlier However he was so impressed with Lobachevsky s work that he saw to it that he was named a corresponding member of the Royal Academy of Sciences Having learned Russian on his own Gauss was able to read Lobachevsky s work in the original Nikolai Ivanovich Lobachevsky Soviet Union 1951 In 1828 at a conference in Berlin Gauss met the young physicist Wilhelm Eduard Weber 1804 1891 He was able to bring him to Göttingen as professor of physics Together they developed a theory of magnetism invented the magnetometer and used the deviation of a compass needle in an electrically generated magnetic field to transmit information between the observatory and the physics institute However they did not develop the economic possibilities of this promising invention of the telegraph In 1836 together with the respected naturalist Alexander von Humboldt 1769 1859 they founded the Göttingen Magnetic Union the first international research society which set as its goal the worldwide investigation of the temporal and spatial changes in the Earth s magnetic field The centimetre gram second CGS system of units developed by Gauss and Weber was officially adopted in 1881 at a scientific congress in Paris there the unit of magnetic induction the gauss was defined 1 text gauss 10 4 text tesla Today one of the methods for determining the Earth s magnetic field is called Gauss s law William Rowan Hamilton Ireland 1943 In 1837 Weber had to leave Göttingen for political reasons seven professors from Göttingen including Gauss s son in law Heinrich Ewald and the brothers Jacob and Wilhelm Grimm were dismissed because they had protested against the abolition of the constitution of the Kingdom of Hanover by the new king Gauss could not bring himself to make a public statement on the matter perhaps because of his conservative and monarchistic leanings In 1839 Gauss drafted a general theory of attractive and repulsive forces potential theory In his later years he wrote a report for the University of Göttingen s widow s bank which included some of the first calculations of the amounts that should be contributed to a retirement account based on actuarial tables and probabilistic considerations Gauss died in 1855 leaving behind a colossal lifetime achievement in many areas of mathematics physics and astronomy In going over his papers and diaries it was discovered that his apparent contempt for the pioneering work of several young mathematicians such as Niels Henrik Abel 1802 1829 and János Bolyai can be explained by the fact that he had made the same discoveries many years earlier but not published his results because they seemed to him incomplete in accord with his motto pauca sed matura little but ripe Richard Dedekind DDR 1981 For example in 1811 he did not consider ripe a theorem on complex functions the main theorem of complex analysis developed fourteen years later by Augustin Cauchy or in 1819 his discovery of the noncommutative multiplication of four dimensional objects that is the quaternions discovered by William Rowan Hamilton 1805 1865 in 1843 Nevertheless he supported some of the students who wrote their doctoral dissertations under his supervision the last of whom were Richard Dedekind 1831 1916 On the Elements of the Theory of Eulerian Integrals and Bernhard Riemann 1826 1866 Hypotheses of the Foundations of Geometry Translated by David Kramer Niels Henrik Abel August 5 1802 April 6 1829 by Heinz Klaus Strick Germany Ayrıntılar Kategori Strick When Niels Henrik Abel was born his homeland of Norway was on the front lines of a European war Both Napoleon and Great Britain s allies had enforced a blockade against the country which was thus cut off from other countries Norway which was only formally still a part of Denmark was occupied by Swedish troops and in 1814 was ceded to the king of Sweden Eventually Norway achieved independence Although Abel s father had a secure position as a Protestant minister Niels Henrik Abel grew up with his six brothers and sisters in difficult economic circumstances Initially he was educated by his father at age 13 he was permitted together with his older brother to attend the cathedral school in the capital Christiania today Oslo At first the transfer to this school failed to represent instructional improvement since the formerly highly respected school had lost its entire faculty to the newly established university Finally the qualified mathematician Bernt Holmboë was engaged who not only recognized the boy s unusual mathematical talent but was able to nurture it He explained to him works of Euler and Newton It seemed to Abel that those mathematicians proofs of the generalized binomial theorem were not sufficiently rigorous and he filled the gaps in their proofs with a precision that was unusual for that time He also extended the statement of the theorem to an arbitrary real exponent When Abel was 18 years old his alcohol addicted father died his unstable mother was unable to care adequately for her younger children It seemed impossible on financial grounds that Abel would be able to continue his education and later study at university seemed out of the question Holmboë managed however to convince some friends to provide money for Abel s education In his last year of secondary school Abel managed to solve or so he believed a hitherto unsolved famous problem of long standing namely that of expressing the solutions of a fifth degree polynomial equation solution methods for equations of degree 4 or less had been known since the Renaissance in terms of the basic arithmetic operations and extraction of roots such an equation is said to be solvable in radicals There was no one in Norway capable of following his argument so it was proposed that he present his ideas to mathematicians at the University of Copenhagen However he noticed that there was an error in his proof And some years he understood why it is in fact impossible to find a general procedure for arbitrary equations of the fifth order At the age of 19 Abel matriculated at the University of Christiania where however there were no courses in higher mathematics Holmboë convinced his former professor Christopher Hansteen who later became known for his research on the Earth s magnetic field to win this unusual talent for science Hansteen took a personal interest in Abel including financing a trip to Copenhagen so that he could discuss his ideas with the Danish mathematician Ferdinand Degen This gave Abel the imp etus to study the so called elliptic integrals For example to determine the arc length of an ellipse whence the name certain integrals must be calculated the function f defined by f x a cdot int 0 x sqrt 1 varepsilon 2 cos 2 t mathrm d t associates an angle x with the arc length f x see the figure Here a and b are the semiaxes of the ellipse and varepsilon sqrt a 2 b 2 a is the numeric eccentricity Such integrals including other related integrals cannot be calculated in an elementary way that is one cannot express in terms of elementary functions a function whose derivative is the expression inside the integral sign Therefore the values of such integrals can be computed only numerically Abel had the idea to investigate the associated inverse function whereby a particular arc length of the ellipse is related to the associated angle thereby creating the theory of elliptic functions He became convinced that to further his work he would have to make personal contact with the greatest mathematicians of his time namely Carl Friedrich Gauss in Göttingen and Adrien Marie Legendre the world expert on elliptic functions and Augustin Louis Cauchy in Paris Over the following two year s he studied the French and German languages intensively and continued his work on mathematical problems In 1824 he published his Mémoire sur les équations algébriques ou on démontre l impossibilité de la résolution de l équation générale du cinquième degré though it appeared in an abridged form to save on the expense of publication and he sent a copy to Gauss who put it aside unread Because of his limited contact with the scientific world Abel was unaware that Paolo Ruffini 1765 1822 had already in 1799 published a paper expounding similar ideas although his proof was marred by several gaps Finally in 1825 he was granted the necessary funds for a journey Carrying letters of introduction he first visited the engineer August Leopold Crelle in Berlin who at the time was preparing to launch a mathematical journal that would be independent of the universities the Journal für die reine und angewandte Mathematik Journal for Pure and Applied Mathematics The first issue of what came to be called Crelle s journal contained seven papers by Abel including the Re cherches sur les fonctions elliptiques and Beweis der Unmöglichkeit algebraische Gleichungen von höheren Graden als dem vierten allgemein aufzu lösen proof of the impossibility of solving general algebraic equations of degree higher than the fourth In the second of these papers he gave a set of criteria for when an equation of higher degree could be solved in radicals In the process of his research he discovered the relationships that had to exist among the solutions arrangements whose order could be reversed whence commutative groups are called abelian groups Since he had been told that Gauss was unapproachable he did not travel to Göttingen journeying instead to Paris via Freiburg Dresden Vienna and Venice However he found there no suitable interlocutor Eventually he submitted to the French Academy of Sciences a paper on the generalization of an addition theorem for elliptic integrals known today as Abel s theorem Legendre and Cauchy were given the paper to review Legendre s age rendered him unequal to the task while Cauchy recoiled at the very idea of having to read it at all Indeed he was so busy with his own publications that he simply put the paper aside and lost track of it When the article was rediscovered after Abel s death he was awarded a posthumous prize from the Academy Abel was disappointed for he knew that a positive judgment

Original URL path: http://mathematics-in-europe.eu/tr/76-enjoy-maths/strick (2013-11-18)

Open archived version from archive - Mathematics In Europe - Anasayfa

1 sort newest Beautiful architectural design physics and mathematics mostly mathematics of course Created by Daniel Piker architectural student based in London Communicated by Steen Markvorsen Denmark http www atractor pt mat puzzle 15 index html A well known puzzle with an interesting twist Communicated by Maria Dedo Italy The top five of January 2011 http www youtube com watch v A ZVCjfWf8 A couple of thought provoking quotes from this video stated by two young students I will have 14 jobs before I am 38 years old Most of those jobs do not exist today Teach me to think to create to analyze to evaluate to apply Mathematics is clearly in demand we must we can engage the students into mathematics and into the future Communicated by Klaus Fink Denmark http www imaginary2008 de Watch the imaginary lemon x 2 z 2 1 y 3 y 3 on display together with several other very well prepared mathematical sculptures Communicated by Martin Raussen Denmark http vimeo com 9953368 A wonderful video about Fibonacci numbers and the golden ratio These topics are common in many popularization activities but here Voronoi tessellations and Delaunay triangulations are also introduced All the information behind the video is given in English and Spanish at the following address http www etereaestudios com docs html nbyn htm intro htm Communicated by Maria Dedo Italy http republicofmath files wordpress com 2009 12 collaboration jpg Collaboration is important and needed and makes the mathematics much more fun Communicated by Steen Markvorsen Denmark http dmf unicatt it paolini penrose youtube html Two fabulous animations with Roger Penrose s darts and kites Communicated by Maria Dedo Italy JavaScript is currently disabled Please enable it for a better experience of Jumi The top five inspirations 2012 Ayrıntılar Kategori Topfive Our Top five inspirations of the month is a collection of web addresses that might be of interest to many of our visitors Here is the 2012 collection The top five of December 2012 Publically oriented web sites The top five of November 2012 Fun and serious stuff concerning knots The top five of October 2012 What is new in www mathematics in europe eu The top five of September 2012 Sets logic and thruth The top five of August 2012 Miscellaneous The top five of July 2012 Miscellaneous The top five of June 2012 The top five of May 2012 The top five of April 2012 The top five of February 2012 The top five of January 2012 The top five of December 2012 The top five of December 2012 have been communicated by John Barrow UK He recommends publically oriented web sites 1 www mathsinthecity com 2 http plus maths org content 3 http sport maths org content 4 http understandinguncertainty org 5 http aperiodical com The top five of November 2012 The top five of November 2012 have been communicated by Franka Miriam Brückler Zagreb Croatia She has collected some interesting links concerned with knot theory 1 Knot Theory http www oglethorpe edu faculty j nardo knots fun htm both fun and serious stuff here 2 Untangling the Mathematics of Knots http www c3 lanl gov mega math workbk knot knot html including a variety of knot activities for use in the classroom popularisation events etc 3 The Knotplot Site http knotplot com lots of knot pictures 4 Mathematics and knots http www popmath org uk exhib knotexhib html an online exhibition of the Division of Mathematics School of Informatics University of Wales Bangor 5 Theory of Tie Knots http www tcm phy cam ac uk ym101 tie aps97tie html not the classical knot theory stuff but new ways to tie a tie including connections to knot theory The top five of October 2012 The top five of October 2012 have been communicated by Ehrhard Behrends Berlin Germany who has collected some new contributions for www mathematics in europe eu from the last weeks 1 Cou can find here the description of many professions where mathematics plays a crucial role 2 The web site of the world mathematical year 2000 is now available again It contains numerous proposals who to present the attractiveness of mathematics to a general public 3 The recent historical article for the mathematical calendar is concerned with Laplace 4 There are new historical articles in our section Stamps and the history of mathematics 5 Jorge Buescu has written an interesting popular article on the Releaux triangle The top five of September 2012 The top five of September 2012 have been communicated by Betul Tanbay who does not want to forget about Logic 1 Let us start with very simple set theory http www youtube com watch v BDYKQtbYxO0 2 Remembering how WE NOUS deal with sets http www youtube com watch v cYsZOjTqK8s 3 Then have a rest with some beautiful Venn diagrams http www newscientist com gallery venn 4 Without ever forgetting main questions does infinity exist http plus maths org content does infinity exist 5 and truth http plus maths org content searching missing truth The top five of August 2012 These top five have been communicated by Steve Humble UK 1 The daily Set puzzle something to start the day www setgame com set puzzle frame htm 2 Enter Michael Barnsley s wonderful world and nothing looks the same again www superfractals com 3 Doodling with Stars investigating pattern http vihart com 4 Unlock the secrets of mathematics each month with the free online maths magazine Plus http plus maths org content Article 5 How to create a Maths Day in your School College www mathsweek ie 2011 puzzles why not run an event yourself The top five of July 2012 These top five have been communicated by Steen Markvorsen Denmark 1 http mathoverflow net A dynamic online workplace where mathematicians ask and answer questions 2 http www youtube com watch v zDZFcDGpL4U feature youtube gdata player A sketch map of changing education paradigms by Sir Ken Robinson Communicated by Hans Romkema Holland and Poul G Hjorth Denmark 3 http www nilesjohnson net hopf html Niles Johnson explains the Hopf fibration Communicated by Lisbeth Fajstrup Denmark 4 http users soe ucsc edu charlie 3body N body orbits Communicated by Hans J Munkholm Denmark 5 http www geometrygames org Jeff Weeks Topology and Geometry Software The top five of June 2012 These top five have been communicated by Sara Santos UK director and co founder of Maths busking 1 Maths in the City Professor Marcus du Sautoy s new project enabling citizens to access Maths Walks in around the world One of the walks in London takes you from Tate Modern across the Millennium Bridge to Saint Paul s Cathedral with a narrative that introduces the visitor to the problem of the Bridges of Königsberg the mathematics and the engineering in bridge design to the Catenary curve and the dome of Saint Paul s Cathedral http www mathsinthecity com 2 The Institute of Mathematics and Its Applications in the UK launched a series of case studies on the impact of mathematical research in the modern world These range from direct applications to pure research that led into improvements in technology such as CAD design http www ima org uk i love maths mathematics matters cfm 3 Mathematics A Beautiful Elsewhere is a unique exhibition created by the Fondation Cartier pour l art contemporain A superb exhibition of mathematics developed by the contemporary art foundation note not a scientific institution Foundation Cartier The work has participation of artist such as David Lynch and the leading mathematicians such as Cédric Villani and Alexandre Grothendieck The Fondation Cartier has opened its doors to the community of mathematicians and invited a number of artists to accompany them They are the artisans and thinkers the explorers and builders of this exhibition http fondation cartier com en art contemporain 26 exhibitions 27 mathematics a beautiful elsewhere 4 Matt Parker Stand up Maths a maths addict comedian Matt started Maths Jam a regular gathering of maths people in cafés bars and pubs around the world see http mathsjam com if you would like to join or organise a gathering in your hometown Matt has a regular feed of puzzles in the UK newspaper The Telegraph http www telegraph co uk education maths reform 9236922 Matt Parkers maths problem page html and a regular feed of maths videos on YouTube http www youtube com standupmaths 5 Maths Busking is the award winning project run by Sara Santos that trains people to do mathematics as street performance It received a Recognition of Distinction from EngageU 2012 the European Competition for University Outreach and last year was awarded the Seed of Science 2011 in Science Communication The website with a video and more info is http mathsbusking com The next training session and performance in Krakow in July will be announced soon The top five of May 2012 Sorry but these proposals by Robin Wilson UK they are concerned with the history of mathematics are so extensive that we must ask you to continue here The top five of April 2012 These top five have been communicated by Raúl Ibanez Torres Bilbao He presents some interesting Spanish websites 1 Divulgamat Virtual Center for the Popularization of Mathematics http www divulgamat net This is the most complete and interesting web page on mathematical culture in Spanish developed by the Royal Spanish Mathematical Society You can find here mathematical challenges history of mathematics virtual expositions art photography history culture and mathematics magic origami cinema literature theatre music publicity math stories publications about the popularization of mathematics texts on line didactic resources applications and more 2 Matematicalia http www matematicalia net Excellent virtual magazine on popularization of mathematics 2005 2011 in Spanish with periodical articles about science communication culture economy education international national society technology puzzles and humour 3 Gaussianos http gaussianos com Blog related with mathematics developed by Miguel Angel Morales The author of the blog writes interesting and well written articles each day This blog is a reference in the world of Spanish blogs related with mathematical culture 4 Carnaval Matemático http carnavaldematematicas bligoo es This is a forum of Spanish mathematical bloggers Tito Eliatron Dixit Tecnología obsoleta Gussianos The blog of Sangakoo etc Each month there is a contest to choose the best article of the collaborators of this mathematical carnival 5 Matemáticas em tu mundo http catedu es matematicas mundo Excellent personal page in Spanish of the professor of mathematics in a secondary school in Zaragoza José María Sorando Mathematics in art cinema sports history poetry literature society and so on The top five of February 2012 These top five have been communicated by Maria Dedó Milano Her leading idea of the month are curves and surfaces mainly from a visual point of view 1 Proposed as the top favourite A gallery of real virtual algebraic surfaces http www dm unito it modelli index html You can find here a beautiful collection of models of algebraic surfaces For each surface in the collection you can compare a photo of a concrete object in the collection of models kept in the University of Torino Italy with a virtual animation of the same surface see for example http www dm unito it modelli modelli mainshell clebsch html the page about the Clebsch surface On the left you can change surface The texts are in Italian but there is much to be enjoyed also by people who do not read Italian 2 A gallery of abstract sculptures http www cs berkeley edu sequin SCULPTS sculpts html The collection includes sculptures of different contemporary artists Bruce Beasley Brent Collins Helaman Ferguson Nat Friedman Bathsheba Grossman George Hart Ken Herrick Robert Longhurst Charles Perry Steve Reinmuth Rinus Roelofs Keuizo Ushio and Carlo Séquin who is the editor of the collection each one exploring through different points of views what a mathematician could call the theme of topological surfaces 3 A gallery of spirals http spiral gallery sytes org A beautiful collection of images both real world images and virtual ones each one showing a spiral from staircases to vegetables 4 A gallery of crochet surfaces http www math uchicago edu mrwright crochet A collection of surfaces mainly Seifert surfaces of knots but also a Klein bottle and the first steps of a horned sphere and all these made with a crochet 5 All the 59 stellated icosahedra http www mathconsult ch showroom icosahedra list graph html A visual index of all the polyhedra which can be obtained by stellation of an icosahedron Each image is clickable to a bigger one with some information Among polyhedral surfaces you can find an analogous visual index for uniform polyhedra here http www mathconsult ch showroom unipoly list graph htm The top five of January 2012 These top five have been communicated by Krzysztof Ciesielski Kraków 1 Jeff Miller Web Pages At the page http jeff560 tripod com Among several web pages there are some very interesting mathematical pages Especially three of them can attract many mathematicians http jeff560 tripod com stamps html Here we may find an enormous collections of mathematical stamps If you want to know what stamps were issued in a particular country or with a particular mathematical topic look at this page http jeff560 tripod com mathword html Here the pages attempt to show the first uses of various words used in mathematics http jeff560 tripod com mathsym html Here these pages show the names of the individuals who first used various common mathematical symbols and the dates the symbols first appeared 2 Estonian Math Competition Olympiad Resources on the Web http www math olympiaadid ut ee eng html index php id links The page contains an enormous and very interesting collection of links to many pages connected with national mathematical competition in different countries Several information and first of all very large collections of problems may be found here 3 Klein bottle http www kleinbottle com Do you want to see a wool Klein bottle hat Several variations of Klein bottles made from glass Cartoons about Klein bottle Etc etc You may find it in this page 4 Films that show how to turn a sphere inside out http www youtube com watch v wO61D9x6lNY feature gv http www youtube com watch v wn qmgOt Js feature related http www youtube com watch v bGiVPj2P19s feature related 5 Famous Escher graphics showing impossible figures may be made from Lego bricks http www bergoiata org fe Escher lego 10 htm A compilation of all links in our top five Ayrıntılar Kategori Topfive Information Popularization Math Help Maths as a profession Enjoy Maths Communication 1 Information 1a What is mathematics 1b Landscape 1c Research 1d History 1e Philosophy 1f Art 1g Music 1h Mathematicians 1i Mathematics in daily life 2 Popularization 2a General Foundations 2b Discrete Math Algebra Numbers Combinatorics 2c Analysis Functions 2d Geometry Topology 2e Applied Mathematics 2f Popular mathematical web pages 2g Other fields 3 Math Help 3a Mathematics in other languages 3b Visualization of Mathematics 3c Teachers 3d Software 3e Learning 4 Maths as a profession 4a Research 4b Education 4c Industry 5 Enjoy maths 5a Puzzles 5b Mathematical magic tricks 5c Games 5d Nature 5e Museums 5f Stamps 5g Videos 5h Architecture 5i Applets 5j Miscellaneous 6 Communication 6a Mathematically oriented blogs A compilation of all links Information Ayrıntılar Kategori Topfive 1a What is mathematics 1b Landscape 1c Research 1d History 1e Philosophy 1f Art 1g Music 1h Mathematicians 1i Mathematics in daily life 1a What is mathematics Mathematical book reviews top five November 2011 proposal 5 Wikipedia Portal Mathematics top five February 2011 proposal 2 top of this page 1b Landscape top of this page 1c Research The Reuleaux triangle top five October 2012 proposal 5 Jeff Week s Topology and Geometry Software top five July 2012 proposal 5 N body orbits top five July 2012 proposal 4 Niles Johnson explains Hopf fibration top five July 2012 proposal 3 The Institute of Mathematics and its Applications top five June 2012 proposal 2 top of this page 1d History Stamps and the history of mathematics top five October 2012 proposal 4 Various links concerned with the history of mathematics were provided in May 2012 Cartography through history top five December 2010 proposal 1 top of this page 1e Philosophy top of this page 1f Art Superposition An audiovisual performance with a strong soundtrack by Ryoji Ikeda top five June 2013 proposal 4 A video inspired by Escher s work top five April 2013 proposal 1 Quaternions embroidered quilts by Gwen Fisher top five January 2013 proposal 5 A 3D analogue of the Mandelbrot set by Daniel White top five January 2013 proposal 4 Escher and the Droste effect top five January 2013 proposal 3 Mathematical Imagery by Jos Leys top five January 2013 proposal 2 An exhibition created by the Fondation Cartier top five June 2012 proposal 3 Mathematics A beautiful elsewhere An exhibition top five June 2012 proposal 3 Mathematics in art cinema sports history poetry literature society by J M Sorando in Spanish top five April 2012 proposal 5 Maths and art top five November 2011 proposal 4 top of this page 1g Music Mathematics in art cinema sports history poetry literature society by J M Sorando in Spanish top five April 2012 proposal 5 top of this page 1h Mathematicians Martin Gardner and Ian Stewart top five August 2013 proposal 5 TED talk by Benoit Mandelbrot top five March 2013 proposal 4 A Mathematical Mystery Tour by Bob Gardner top five March 2013 proposal 2 Gödel exhibition in Vienna top five October 2011 proposal 2 Terry Tao s weblog top five August 2011 proposal 2 Timothy Gower s weblog top five August 2011 proposal 1 Benoit Mandelbrot in memoriam top five November 2010 proposal 4 Oberwolfbach gallery of mathematicians top five October 2010 proposal 5 top of this page 1i Mathematics in daily life Poetic Spirals top five June 2011 proposal

Original URL path: http://mathematics-in-europe.eu/tr/77-enjoy-maths/topfive (2013-11-18)

Open archived version from archive - Mathematics In Europe - Anasayfa

think that these three random processes have resulted in a chaotic mix of the cards the cards arranged totally at random And so it appears at first glance However a remarkable phenomenon is at work It turns out that cards 1 and 2 have different colors as do cards 3 and 4 5 and 6 and so on As the magician you may now place the pack of cards under a cloth and murmur mystical incantations and then pull out pairs of oppositely colored cards as if by magic even though you are actually just drawing the cards from top to bottom See Figure 3 Figure 3 and now present the cards a pair at a time The mathematics behind this is interesting The fact that the cards are arranged pairwise with different colors after the three random events can be proved with combinatorial methods In this connection mathematicians speak of an invariant The magician Gilbreath who invented the trick at the beginning of the previous century seems to have discovered it by trial and error A Variant of the Trick For those who would like to add this trick to their repertoire here is a variant Recall that the original goes according to the following outline Prepare the pack of cards even number of cards of alternating colors Cut the deck and then shuffle the cards Cut the deck at a point where there are two cards of the same color and reassemble the deck Then each pair cards 1 and 2 cards 3 and 4 etc contains two cards of opposite colors And now the variant The deck of cards is prepared just as in the original version and it is again cut Warning This time you must somehow inform yourself as to whether the cards on the bottoms of the two halves are of the same color or different colors This could be done for example as the cards are handed to the shuffler The next step is again as previously the two halves are shuffled together And that is it you do not need to cut the pack again The advantage over the first variant is that you don t need to have someone look at the cut deck to see where two cards of the same color reside Thus no one will get the idea that the colors red and black are more regularly distributed than would be the case in a well mixed pack of cards There are now two cases Case 1 is that the two bottom cards had different colors Then no adjustment is necessary It is guaranteed that every pair cards 1 and 2 cards 3 and 4 and so on are of opposite colors The second case is that the two cards on the bottom were both red or both black Now things are a bit more complicated While you are muttering your magic formulas move the top card to the bottom of the deck Then again all pairs will contain one red card and one black Of course you don t have to move the top card to the bottom Instead your first pair should consist of the top and bottom cards Thereafter things proceed as before Good luck with your magic And where is the mathematics in all of this It guarantees that the trick will always work It can be proved that the cards end up as described here However the rather complex theory necessary for the proof is beyond the scope of this column This is an article from the book Five minute mathematics by Ehrhard Behrends which was published in 2008 by the American Mathematical Society AMS It is reproduced here with the kind permission of the AMS JavaScript is currently disabled Please enable it for a better experience of Jumi Magical Invariants Ayrıntılar Kategori Recreational mathematics By Ehrhard Behrends Berlin What remains unchanging On what can one rely For centuries mathematicians have searched for invariants that is to put it simply quantities whose values do not change with respect to some operation under consideration As an example let us consider a deck of cards When you shuffle the deck there are certain invariants in play Certainly the number of cards is unchanged as are the numbers of jacks queens and so on The situation is different when one does not allow shuffling but only repeated cutting of the pack and putting the bottom half on top of the top half Then the relative order of the cards remains unchanged If the ace of spaces was to be found three cards down from the queen of hearts it will remain so Of course we have to interpret the words down from If the queen of hearts happens to be the bottom card in the deck then the ace of spades will be three cards down from the top That is down from is to be understood in the sense that when you reach the bottom of the deck you continue from the top One can make use of this invariant in a little magic trick Remove the kings and queens from the pack and lay them out in a row as shown in Figure The key here is to make the distance between cards of the same suit king of spades and queen of spades king of clubs and queen of clubs and so on exactly four If you flash this arranged hand of cards in front of your audience it appears to be more or less randomly arranged No one will suspect your skulduggery and if you cut the pack a few times Figure everyone will believe that the cards have been thoroughly mixed Figure 1 The prepared cards But you are operating with the knowledge that the distance between cards is an invariant four cards below the first card is its partner It is therefore easy to produce a pair from under a cloth or under the table of course making it seem that you are struggling mightily You can repeat this process and produce a second pair though this time the distance between them is three and then a third pair separated by two and finally the last pair This trick relies on the existence of some kind of order amidst seeming chaos In mathematics the search for the unchanging has become a sort of leitmotiv in research Once a set of permissible transformations has been described a systematic search begins for quantities that remain unchanged under those transformations This idea has been of particular importance as a unifying principle in many branches of geometry It was proposed in 1872 by the mathematician Felix Klein and has had great influence over research ever since Figure 1 2 The cards after being cut The Back Story The Distance Modulo the Number of Cards Is Invariant Using the notion of modular arithmetic introduced in another chapter of this book we can formulate the principle that underlies the trick somewhat more mathematically If n cards are stacked and two of these cards are in positions a and b counted from the top then b a modulo n is an invariant after cutting the pack any number of times the difference between the positional values has not changed modulo n To see this one has to use modular calculations for negative numbers as well as positive but that is really no problem After all as everyone knows the day of the week seven days ago is the same as the day today and the day 13 days ago was Tuesday if today is Monday Mathematically 13 modulo 7 is equal to 1 One must be aware of this subtlety in order to interpret the invariant relationship presented above correctly An example In the example that follows we are going to use the fact that 7 modulo 10 is equal to 3 In a pack of ten cards the ace of hearts and jack of clubs are in positions 2 and 5 The difference is 3 The pack is cut at position 2 Now the ace of hearts is at position 10 the bottom of the pack and the jack has moved up to position 3 The difference position of the second card minus that of the first is thus 3 10 7 and modulo 10 this is the same number 3 as before We Draw on an Extensible Surface Only a few mathematical invariants are suitable for magic tricks Their great significance is that invariants separate the essential in a theory from the inessential To show this via a somewhat unconventional example we require a drawing surface made out of some stretchable material On our surface we draw a figure a triangle a circle a collection of rectangles whatever Now the surface is distorted we pull it and compress it in any way that strikes our fancy Our drawing will change significantly A small circle can become a large circle a right angle can become obtuse or acute If the original figure had the property that one could connect any two points with a curve contained entirely within the figure which is the case for a circle or triangle but not a collection of rectangles then one can still do so after the figure has been altered connectivity is an invariant under distortion This is an article from the book Five minute mathematics by Ehrhard Behrends which was published in 2008 by the American Mathematical Society AMS It is reproduced here with the kind permission of the AMS JavaScript is currently disabled Please enable it for a better experience of Jumi A Beer Trick Ayrıntılar Kategori Recreational mathematics By Franka Brückler Zagreb Instructions for the reader The trick is performed by a mathe magician with a participator In the following dialogue the mathemagician s text is printed in Roman and the participator s in Italics If you do not know the trick I encourage you to take the participators part and perform his actions as you read down the dialogue A calculating conversation What is your favourite drink Ginger Ale dear reader please insert your favourite drink here How many of your favourite drinks do you approximately drink every day Don t tell me but remember the number or write it down Hmm thinking 4 you the reader are free to choose your number it should be a nonnegative integer here and perform the following calculations with it Today it is very hot so you are very thirsty You drink twice your average so multiply your number with 2 2 4 8 Did it What next You are walking down the street and meet a friend you haven t seen for a long time He invites you to go with him to a bar and you accept You have much to talk about and as the time passes you have drunk 5 more of your favourite drinks so add 5 to your last number 8 5 13 Err I think this it too much for me Don t worry I don t expect you to drink that much At least not right away As the obtained number is obviously much above your daily average we ll take it as an approximation of your weekly consumption In fact I m more interested in finding estimates of how many of their favourite drinks people drink yearly The year has 52 weeks but as I m only interested in estimates it is enough if you multiply your last number with 50 50 13 650 Did it Do you wan t to know the result No not yet Did you celebrate your birthday this year already If no add 1760 to your last number otherwise add 1761 No 650 1760 2410 Now for your last exercise in arithmetics subtract the year of your birth from the previous result The full four digit year please Tell me the result when you re done I m born in 1971 so 2410 1971 439 My final result is 439 You are 39 the last two digits years old and drink 4 the number obtained removing the last two digits of the result of your favourite drinks daily How the mathematician becomes a mind reader The explanation of the trick is simple algebra of the level everybody learns in school If you the reader are a mathematics teacher you can easily make your students discover the following explanation or parts of it e g how the number to be added before subtracting the year of the birth depends on the calendar year in which the trick is performed themselves The explanation is as follows If m is the daily number of drinks the first part of the trick calculates first 2 m then 2 m 5 and then 2 m 5 50 100 m 250 By adding a number k and subtracting the birth year n we obtain the final score 100 m 250 k n Since multiplying by 100 moves m two digits to the left many simple mathemagical tricks are based on this simple fact and use sequential calculations to hide the principle in order to have the last two digits represent the age of the person we must have age 250 k n Since age this year n if the birthday already passed we obtain k this year 250 so in 2011 k 1761 If the birthday hasn t passed yet obviously k has to be one less 1760 in year 2011 Important note If you will present this trick in 2012 you will have to work with the number 1762 and 1761 instead of 1761 and 1760 in 2013 with 1763 and 1762 etc About the trick The trick presented above is one of my favourite mathemagical tricks since the public always loves it it is very simple and quick to perform it works automatically and for persons not used to think mathematically it seems that the performer is a mind reader The trick is not mine in origin I picked it up from the Internet some ten years ago as a result of a search for math beer and the above suggestion for performance is based on that original web site which I was unable to find again Still the trick without the styling can be found on the web e g at http beerexpedition blogspot com 2009 07 your age by beer math html Finally a word of caution Because of the age disclosure it is advisable when performing it grown ups to choose a male participator or to make the age disclosure as private as possible e g not by asking for the final score but just telling how to find the age and number of drinks from it The four ducks trick Ayrıntılar Kategori Recreational mathematics Franka Miriam Brueckler Dept of Mathematics Univ of Zagreb Croatia The described trick is a ducky version of the trick known as Yates Four Divination which is described in Martin Gardner s book Mathematics Magic and Mistery Dover Publ New York 1956 Move the ducks There are four different rubber ducks on the table The participator is to select the one in his favourite colour and name it Then the mathemagician turns his back to the table and gives the following instruction Make five switches of two ducks In every switch your chosen duck must be involved and it may only be switched with a duck adjacent to it Tell me when you have done the five switches Say the participator selected the pink duck He could perform the five switches as follows Switch pink and blue duck Switch pink and yellow duck Switch pink and yellow duck again Switch pink and blue duck Switch pink and green duck After the participator announces that he hase done the five switches the mathemagician still with his back turned to the table asks him to remove a duck at one of the two ends The mathemagician gives a precise instruction Please remove the duck at your left or right depending on which duck was chosen in the beginning In our example the mathemagician would instruct the participator to remove the duck that is on the left of the picture This would leave the participator with the three non yellow ducks In the last part of the trick the mathemagician gives the following instructions and the participator follows them Now please make one more switch of your duck with an adjacent one In our case this would mean switching the pink and green duck Now please remove the duck on the left or right end In our example the removal of the left would be asked and this leaves us without the blue duck And finally please remove the right or left duck This should leave you with your chosen duck In our example the mathemagician would ask to remove the right green duck and the chosen pink duck is left all alone on the table The 4 Ducks manual There are two principal questions to think about when trying to explain and learn the trick How does the mathemagician know how to instruct the participator in the removal of the ducks And what about using other numbers of switches besides 5 In fact consideration of the second question leads to the answer to the first one If you would like to work it out yourself try what happens if 1 2 3 4 switches are required Think about possible final configurations of the ducks in each of the cases As a hint we give you the following number the starting positions of the ducks with 1 2 3 and 4 Now if you do not want the trick disclosed to you before you have tried to find out about it on your own please do not read further All other readers will find the explanation in the following paragraphs If the starting positions are numbered 1 to 4 say 1 for the starting position of the yellow duck in the first of the pictures above then if the chosen duck started from an even position after 5 switches it will end up in an odd position and vice versa In our example the chosen pink duck started from position 3 so after 5 switches it must end up in position 2 or 4 This enables the performer to eliminate one of the two ducks at the ends If the starting position was odd like in our example the performer can be sure that the duck is now not on position 1 and can give the instruction to remove the duck at the corresponding end After this the chosen duck is sure not to be in the middle of the three ducks left on a table and consequently another switch with an in fact the adjacent duck brings it to the middle position Now it is easy to instruct the participator to remove the flank ducks and end up with the chosen duck Finally note that the performance would be exactly the same if instead of 5 any other odd number of switches was used If one uses an even number of switches then the chosen duck will end in a position of same parity as its starting position even if starting from an even one and odd if starting from an odd one and it is easy to adapt the performance to this case There is more maths behind the trick than meets the eye Although the trick can be explained on a pure logical basis like above the explanation can be connected to a relatively advanced mathematical notion of odd and even permutations A permutation is a rearrangement of a number of things in our example every rearrangement of the four ducks is one permutation of the ducks It relates to the relative positions of the things and not to the things itself any rearrangement of the numbers 1 2 3 and 4 can be related to a rearrangement of the four ducks or any other four items if we number their starting positions e g the permutation 2 4 1 3 would in the duck context meant that the duck that was on position 2 in the beginning is now on position 1 the one who started from position 4 is now on position 2 the one who started from position 1 is now on position 3 and the one who started from position 3 is now on position 4 For those who are interested A formal definition of a permutation of a permutation of a usually finite set is that it is a bijection of the set onto itself Permutations can be classified as even and odd By definition an even odd permutation is one that is obtainable by an even odd number of transpositions A transposition is a permutation that interchanges two objects and does not move the others so each of our duck switches is a transposition on our set of ducks Consequently if the participator is asked to switch the ducks 5 times he is asked to perform an odd permutation There is however a condition on the transpositions performed one is allowed only to interchange neighbouring ducks Since two adjacent positions are obviously of different parity it is almost obvious that an even permutation composed of transpositions of neighboring objects does not change the parity of the position of the chosen object and an odd one always changes it Permutations are studied both in combinatorics and abstract algebra and parity is one of the properties helping in their study Parity really provides a classification of permutations although the same permutation final distribution of objects can be achieved by different sequences of transpositions try to find another sequence of switches that rearranges the ducks from the starting configuration to the last one before removing any ducks in all possible sequences there will always be involved an odd number or always an even number of transpositions In other words if a permutation is obtained by an odd number of transpositions it can never be obtained by any sequence of an even number of transpositions nor vice versa In our example this means that no even number of duck switches not even if we allow switching ducks that are not adjacent will transform the starting configuration to the last one before removing any ducks This is known as the parity theorem for permutations One of the best known applications of using this classification is the solution to the famous Sam Loyd s 15 puzzle Two rows of pebbles Ayrıntılar Kategori Recreational mathematics Franka Miriam Brueckler Dept of Mathematics Univ of Zagreb Croatia For this trick you need a number of smaller items e g pebbles or matchsticks coins marbles toothpicks Each of them we illustrate by a The performer mathemagician asks a spectator to choose any odd number of pebbles to perform the trick but the chosen number of pebbles is not disclosed to the mathemagician who during the whole trick remains turned so that he can t see the table As soon as the spectator has chosen the number of pebbles to work with and has removed the rest the mathemagician asks him to arrange the pebbles in two rows so that the bottom row contains one pebble more than the top one For example if the spectator chose to work with 17 pebbles he would arrange them like this Now the mathemagician asks the spectator to name a positive integer number smaller than the number of pebbles in the top row In our example the spectator would choose a number smaller than 8 say 6 The mathemagician instructs him to remove that many pebbles from the top row In our example this leaves the spectator with Then the mathemagicians asks the spectator to remove from the bottom row as many pebbles as there are left in the top row In our example the spectator removes 2 pebbles from the bottom row and obtains situation Finally the mathemagician instructs the spectator to remove all the pebbles that still remain in the first row After that the mathemagician guesses correctly of course the remaining number of pebbles In the described example the last step would be removing the three pebbles from the top row and the mathemagician would announce that there are 7 pebbles left The background of the trick is simple basic algebra We encourage the reader to try to work it out by him or herself For those who prefer to have the full description of the trick or want to check if their solution is correct the following paragraph gives the full explanation First note that any odd number of pebbles can be divided in two rows differing by 1 in the number of pebbles So if we denote by n the number of pebbles in the top row at the beginning of the trick the bottom row contains n 1 pebbles in the described example n 8 Denote by m the number named by the spectator in our example m 6 The first instruction leaves the spectator with n m pebbles in the top row and the bottom one still contains n 1 of them Then the spectator removes as many pebbles from the bottom row as there are in the top row i e removes n m pebbles from the bottom row so the bottom row now contains n 1 n m m 1 pebbles Eliminating the top row one eliminates the dependence of the number of pebbles on the table on the first chosen number unknown to the mathemagician and leaves m 1 pebbles on the table Since the mathemagician knows the value of m it is easy for him to guess the number of pebbles remaining on the table That s all folks This trick is described without explanation of the mathematical background under the name Coin rows in Oliver Ho s book Amazing Math Magic Sterling Publ Co New York 2002 Gergonnes trick Ayrıntılar Kategori Recreational mathematics Franka Miriam Brueckler Department of Mathematics University of Zagreb Croatia Gergonne s trick is probably the best known of the mathematically based magic tricks and is known in several variations It was first analysed and generalised by the 19 th century French mathematician Joseph Diaz Gergonne and is described including or not various extensions in many recreational mathematics books e g in Gardner s Mathematics Magic and Mistery Here we present the basic version of the trick using 27 cards The 27 cards are dealt face open in three columns with 9 cards in each The performer asks a spectator to choose a card but to name only its column Say the spectator has chosen the card the top card in the rightmost column in the picture above he names the rightmost column as the chosen one The performer collects the cards and then deals them out again In the described case the cards will be rearranged as shown in the next picture The spectator is asked to name the column with his card again in our example case this would now be the leftmost column The procedure performer collects the cards deals them out again and asks the spectator for the column with his card is repeated In our case the third layout of cards would look like this Now the spectator would again name the leftmost column Once more the performer collects and deals the cards but now he does not ask questions Instead he uses his mathemagical powers concentrates and reveals that the chosen card is now in the center of the layout In fact the previous procedure will always bring the chosen card to the central position provided the performer takes care of the following rules always pick up the cards by columns put the indicated column between the other two and deal the cards out by rows Why does it work In this basic version the secret is a basic principle of elimination indicating the column reduces the number of possible cards to 9 by putting this column in the center and dealing out by rows these 9 cards will be the central ones A second indication of the column thus now reduces the number of possible cards to 3 which land in the central row after the second round of collecting and dealing The final question is basically not needed the card is now in the center of the indicated column and the repeated procedure will bring it to the center We finish this short article by a simple question for the reader If you start with 3 n cards how many repetitions of the procedure collect and deal are needed to bring the chosen card to the central position The clairvoyant mathe magician Ayrıntılar Kategori Recreational mathematics Franka Miriam Brueckler Department of Mathematics University of Zagreb Croatia I suppose that a pack of cards is the most important prop for a populariser of mathematics Or a mathematics teacher Or a mathematics enthusiast Or Playing cards can be used to teach numbers obviously introduce variables or what else is a joker learn basic logic skills through various card games to explain some basic probability principles I guess there is no need to explain this In this article we shall describe a magical trick with cards founded on elementary algebra There are several versions of this trick but the basic one as described in Martin Gardner s Mathematics Magic and Mistery under the name A Baffling Prediction works best in public at least to this article author s experience Let me the author be your performer and you the reader the spectator willing to participate in the trick I give you a deck of cards a standard one with 52 cards to shuffle I ask you to deal any 12 cards you choose face up and I will hold the rest of the pack while you arrange them Twelve cards are dealt face open on the table Please select four of them and leave them on the table and return the other eight to me I say You do as I say Now let us agree on the card values If you want to suggest some please do No O K say we use the standard values cards with numerals printed on them have the corresponding value aces have value 1 and jacks queens and kings have value 10 Is this O K with you Yes Fine Now I return the pack of cards to you Four cards are left on the table Well you ask Well I say now deal from the pack onto each of the face up cards as many cards from the pack as is the difference between 10 and the card value Face up you ask I don t care just take care to count correctly is my answer So you deal 7 cards on the three of hearts one card on the nine of hearts 6 cards on the four of diamonds and no cards on the king of clubs And you still have some cards left in your hand Onto each card as many cards are dealt as is the value of the difference between 10 and the card value Now I will concentrate is my line and you see me closing my eyes putting my thinking magician s hat on Yes I have it You have what you ask I think I am able to see the twenty sixth card in the pack in your hand Please find it Holding the pack face up or face down Down if you please And you do as you have been told to do And I say I see something red and a number with two digits and a man with a hat he looks somehow sleepy It s Robert Mitchum i e the 10 of diamonds isn t it And you are surprised or at least you act as if you were because I am right The performer knows the 26th card if the cards have been chosen as shown in the previous pictures Now dear reader you ask yourself what has happened First of all remember the time you have returned the pack of cards to me Yes The 10 of diamonds with the picture of actor Robert Mitchum was at the bottom and I had plenty of time to see that while you were choosing the four cards But this it the only unfair thing I ve done I m a mathematician and I don t cheat I let maths work for me Does the previous information help you to find how I ve known which number to name if I want to find the 10 of diamonds No Then read on We had 52 cards in the beginning thus you have returned to me 52 12 40 cards in the first step So the 10 of diamonds was the 40th card from the top in that pack You have returned 8 cards to me I have put them below the pack so the Robert Mitchum is still at position 40 Everything clear You still don t see But your open cards were the 3 of hearts 9 of hearts 4 of diamonds king of clubs 26 in total value Does this help No Then I have to teach you some algebra You have four cards open with values a b c and d According to my instructions you place onto them 10 a 10 b 10 c 10 d cards i e 40 a b c d That many cards are still in the pack and the 10 of diamonds or whatever card I ve seen when you returned the pack to me is at the bottom Independently on how we agreed on the values a b c d this card is now on the position a b c d from the top so all I had to do is to add the agreed upon values of the four open cards to tell you where the queen of hearts is The Three Ducks Trick Ayrıntılar Kategori Recreational mathematics Franka Miriam Brueckler Dept of Mathematics Univ of Zagreb Croatia The described trick is a ducky version of the trick known as Hummer s 3 Object Divination which is described in Martin Gardner s book Mathematics Magic and Mistery Dover Publ New York 1956 Duck Switches There are three different rubber ducks on the table The mathemagician and the participator agree on naming the positions e g the position of the yellow duck above they name position 1 the middle position they name 2 and the third one they name 3 After that the participator is to select one duck but not name it Then the mathemagician turns his back to the table and gives the following instruction First switch the two ducks you haven t chosen and do not tell me the positions involved After that you can switch any two ducks as many times as you want but for all these other switches you tell me the positions involved After that the mathemagician turns around and can instanteneously tell which duck the participator has chosen in the beginning Say the participator selected the yellow duck Then in the first switch he would change the positions of the pink and blue duck After that he continues switching but with naming the positions say he switches now the pink and yellow duck In this case he says that he has switched positions 1 and 2 Then he decides to switch the pink and yellow duck again saying the switch involved positions 1 and 2 When switching the pink and blue duck he says that he switches 2 and 3 Say that the participator decides to stop here The mathematician turns around and says You chose the yellow duck How it works The principle of this trick is simple logic All the

Original URL path: http://mathematics-in-europe.eu/tr/78-enjoy-maths/recreational-mathematics (2013-11-18)

Open archived version from archive - Mathematics In Europe - Anasayfa

completed For the corresponding question in three dimensions that is for three dimensional space the problem seemed hopeless In order to explain the Poincaré problem we need to introduce a bit of terminology namely the notion of simple connectedness Imagine your living quarters with all the furnishings removed the front door closed and all the internal doors removed from their hinges Now take a long thread pull it around through the various rooms in any way that you like and then tie the two ends together If you pull on the loop either the whole thread will glide into your hand or else it will get hung up somewhere because your dwelling allows for a circular path through the rooms as in the right hand picture in Figure Figure 1 2 Is your home simply connected So here s the definition A region of space is said to be simply connected if as in the left hand picture of Figure 1 2 it is impossible for a thread as described above to get hung up This definition can also be applied to two dimensional surfaces The surface of a sphere is certainly simply connected while the surface of a life preserver is not Poincaré conjectured that there is only one region in space that is simply connected and in a certain technical sense is not too large That was around the year 1900 Since then enormous progress has been made in our understanding of space yet the problem remained unsolved This situation was most unsatisfactory since in the meanwhile problems in other areas of mathematics of apparently much greater difficulty had been solved However it seems now that a solution is imminent This cautious formulation is necessary because the proof strategy proposed by the Russian mathematician Grigori Perelman has not

Original URL path: http://mathematics-in-europe.eu/tr/79-supply/five-minutes (2013-11-18)

Open archived version from archive - Mathematics In Europe - Startside

Juridisk ansvarlige Søg The European Mathematical Society Our Sponsor Munich RE National activities Detaljer Kategori National activities The popularization of mathematics has played a special role since the turn of the millennium At that time an International Year of Mathematics was proclaimed and a variety of activities in various countries took place Exhibitions posters newspaper articles and so on Even the organizers of the International Congresses of Mathematicians have been

Original URL path: http://mathematics-in-europe.eu/da/50-popularization/national-activities (2013-11-18)

Open archived version from archive - Mathematics In Europe - The Translation Project

appearance Article Number Title Original Article by Author Media Partner Link Translations available in 2013 III Dal mappamondo all atlante Siliva Benvenuti xlatangente Original Article Link Print Italian Original Portuguese 2013 II The Tower of Hanoi Where maths meets psychology Marianne Freiberger Plus magazine Original Article Link English Original Italian German 2013 I Gnash a flat torus Vincent Borelli images des Maths Original Article Link French Original English Italian Portuguese The Network Currently the following partners of content providers and content publishers agreed to offer articles for translations and or publish translated articles The list is in alphabetical order Divulgamat Spain images des Maths France IMAGINARY International Mathematics in Europe Europe Plus magazine UK xlatangente Italy Please contact us at translation project at mathematics in europe eu if you are interested to join the media network The Translators and Translations We are establishing a pool of mathematicians and translators to work with us to translate the articles into many European languages and proof read them Please let us know if you are interested to join the project We usually pay a lumpsum amount per article and offer a very flexible working environment Please contact us at translations translation project at

Original URL path: http://mathematics-in-europe.eu/da/18-information/996-the-translation-project (2013-11-18)

Open archived version from archive - Mathematics In Europe - Bridges Conference 2013

since 1998 Here is the mission statement The International annual conference of Bridges Mathematical Connections in Art Music and Science was created in 1998 and is conducted annually It has provided a remarkable model of how seemingly unrelated and even antipodal disciplines such as mathematics and art can be crossed During the conference practicing mathematicians scientists artists educators musicians writers computer scientists sculptures dancers weavers and model builders have come

Original URL path: http://mathematics-in-europe.eu/da/nyheder/10-frontpage/news/1000-bridges-conference-2013 (2013-11-18)

Open archived version from archive - Mathematics In Europe - Gaps between prime numbers

Society Our Sponsor Munich RE Gaps between prime numbers Detaljer Kategori News The prime numbers 2 3 5 7 11 have attracted the attention of mathematicians since centuries How small can the gaps between very large primes be Are there e g infinitely many examples where both p and p 2 are primes so called twin primes This problem is far from being solved but recently it has been proved

Original URL path: http://mathematics-in-europe.eu/da/nyheder/10-frontpage/news/997-gaps-between-prime-numbers (2013-11-18)

Open archived version from archive