121-412: Johann Peter Gustav Lejeune Dirichlet ( / ˌ d ɪər ɪ ˈ k l eɪ / ; German: [ləˈʒœn diʁiˈkleː] ; 13 February 1805 – 5 May 1859) was a German mathematician . In number theory , he proved special cases of Fermat's last theorem and created analytic number theory . In analysis , he advanced the theory of Fourier series and was one of the first to give the modern formal definition of
242-588: A brain hemorrhage . Jacob von Staehlin [ de ] wrote a short obituary for the Russian Academy of Sciences and Russian mathematician Nicolas Fuss , one of Euler's disciples, wrote a more detailed eulogy, which he delivered at a memorial meeting. In his eulogy for the French Academy , French mathematician and philosopher Marquis de Condorcet , wrote: il cessa de calculer et de vivre — ... he ceased to calculate and to live. Euler
363-522: A convex polyhedron , and hence of a planar graph . The constant in this formula is now known as the Euler characteristic for the graph (or other mathematical object), and is related to the genus of the object. The study and generalization of this formula, specifically by Cauchy and L'Huilier , is at the origin of topology . Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of
484-502: A function . In mathematical physics, he studied potential theory , boundary-value problems , and heat diffusion , and hydrodynamics . Although his surname is Lejeune Dirichlet, he is commonly referred to by his mononym Dirichlet , in particular for results named after him. Gustav Lejeune Dirichlet was born on 13 February 1805 in Düren , a town on the left bank of the Rhine which at
605-412: A book he kept close for his entire life. In 1823 he was recommended to General Maximilien Foy , who hired him as a private tutor to teach his children German , the wage finally allowing Dirichlet to become independent from his parents' financial support. His first original research, comprising part of a proof of Fermat's Last Theorem for the case n = 5 , brought him immediate fame, being
726-416: A candidate for its presidency by Jean le Rond d'Alembert , Frederick II named himself as its president. The Prussian king had a large circle of intellectuals in his court, and he found the mathematician unsophisticated and ill-informed on matters beyond numbers and figures. Euler was a simple, devoutly religious man who never questioned the existing social order or conventional beliefs. He was, in many ways,
847-645: A career in law. As Germany provided little opportunity to study higher mathematics at the time, with only Gauss at the University of Göttingen who was nominally a professor of astronomy and anyway disliked teaching, Dirichlet decided to go to Paris in May 1822. There he attended classes at the Collège de France and at the University of Paris , learning mathematics from Hachette among others, while undertaking private study of Gauss's Disquisitiones Arithmeticae ,
968-477: A couple of papers in 1838 and 1839, he proved the first class number formula for quadratic forms (later refined by his student Kronecker). The formula, which Jacobi called a result "touching the utmost of human acumen", opened the way for similar results regarding more general number fields . Based on his research of the structure of the unit group of quadratic fields , he proved the Dirichlet unit theorem ,
1089-575: A dissertation that compared the philosophies of René Descartes and Isaac Newton . Afterwards, he enrolled in the theological faculty of the University of Basel. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono with which he unsuccessfully attempted to obtain a position at the University of Basel. In 1727, he entered the Paris Academy prize competition (offered annually and later biennially by
1210-477: A financial economist might study the structural reasons why a company may have a certain share price , a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock ( see: Valuation of options ; Financial modeling ). According to the Dictionary of Occupational Titles occupations in mathematics include
1331-477: A fundamental result in algebraic number theory . He first used the pigeonhole principle , a basic counting argument, in the proof of a theorem in diophantine approximation , later named after him Dirichlet's approximation theorem . He published important contributions to Fermat's Last Theorem , for which he proved the cases n = 5 and n = 14 , and to the biquadratic reciprocity law . The Dirichlet divisor problem , for which he found
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#17327662317961452-795: A house in Charlottenburg , in which he lived with his family and widowed mother. Euler became the tutor for Friederike Charlotte of Brandenburg-Schwedt , the Princess of Anhalt-Dessau and Frederick's niece. He wrote over 200 letters to her in the early 1760s, which were later compiled into a volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess . This work contained Euler's exposition on various subjects pertaining to physics and mathematics and offered valuable insights into Euler's personality and religious beliefs. It
1573-400: A manner which will help ensure that the plans are maintained on a sound financial basis. As another example, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while
1694-656: A medic in the Russian Navy . The academy at Saint Petersburg, established by Peter the Great , was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made especially attractive to foreign scholars like Euler. The academy's benefactress, Catherine I , who had continued the progressive policies of her late husband, died before Euler's arrival to Saint Petersburg. The Russian conservative nobility then gained power upon
1815-608: A musician and perhaps been leader of a group. Next to Felix and Fanny, I could not aspire to any recognition. In 1832 she married Dirichlet, who was introduced to the Mendelssohn family by Alexander von Humboldt . In 1833 their first son, Walter, was born. She died in Göttingen in 1858. As soon as he came to Berlin, Dirichlet applied to lecture at the University of Berlin , and the Education Minister approved
1936-469: A new field of study, analytic number theory . In breaking ground for this new field, Euler created the theory of hypergeometric series , q-series , hyperbolic trigonometric functions , and the analytic theory of continued fractions . For example, he proved the infinitude of primes using the divergence of the harmonic series , and he used analytic methods to gain some understanding of the way prime numbers are distributed. Euler's work in this area led to
2057-788: A political dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked and scraping off her skin with clamshells (some say roofing tiles). Science and mathematics in the Islamic world during the Middle Ages followed various models and modes of funding varied based primarily on scholars. It was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. Funding for translation of scientific texts in other languages
2178-404: A quarter of the combined output in mathematics, physics, mechanics, astronomy, and navigation in the 18th century. Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function and was the first to write f ( x ) to denote the function f applied to the argument x . He also introduced
2299-462: A result otherwise known as the Euclid–Euler theorem . Euler also conjectured the law of quadratic reciprocity . The concept is regarded as a fundamental theorem within number theory, and his ideas paved the way for the work of Carl Friedrich Gauss , particularly Disquisitiones Arithmeticae . By 1772 Euler had proved that 2 − 1 = 2,147,483,647 is a Mersenne prime. It may have remained
2420-500: A rigorous proof of convergence. The memoir pointed out Cauchy's mistake and introduced Dirichlet's test for the convergence of series. It also introduced the Dirichlet function as an example of a function that is not integrable (the definite integral was still a developing topic at the time) and, in the proof of the theorem for the Fourier series, introduced the Dirichlet kernel and the Dirichlet integral . Dirichlet also studied
2541-616: A royal pensioner, their friendship becoming even closer. In 1846, when the Heidelberg University tried to recruit Dirichlet, Jacobi provided von Humboldt the needed support to obtain a doubling of Dirichlet's pay at the university in order to keep him in Berlin; however, even then he was not paid a full professor wage and could not leave the Military Academy. Holding liberal views, Dirichlet and his family supported
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#17327662317962662-460: A solution to several unsolved problems in number theory and analysis, including the famous Basel problem . Euler has also been credited for discovering that the sum of the numbers of vertices and faces minus the number of edges of a polyhedron equals 2, a number now commonly known as the Euler characteristic . In the field of physics, Euler reformulated Newton 's laws of physics into new laws in his two-volume work Mechanica to better explain
2783-593: A strong connection to the academy in St. Petersburg and also published 109 papers in Russia. He also assisted students from the St. Petersburg academy and at times accommodated Russian students in his house in Berlin. In 1760, with the Seven Years' War raging, Euler's farm in Charlottenburg was sacked by advancing Russian troops. Upon learning of this event, General Ivan Petrovich Saltykov paid compensation for
2904-501: A way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis . He invented the calculus of variations and formulated the Euler–Lagrange equation for reducing optimization problems in this area to the solution of differential equations . Euler pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced
3025-467: A younger brother, Johann Heinrich. Soon after the birth of Leonhard, the Euler family moved from Basel to the town of Riehen , Switzerland, where his father became pastor in the local church and Leonhard spent most of his childhood. From a young age, Euler received schooling in mathematics from his father, who had taken courses from Jacob Bernoulli some years earlier at the University of Basel . Around
3146-420: Is mathematics that studies entirely abstract concepts . From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics , and at variance with the trend towards meeting the needs of navigation , astronomy , physics , economics , engineering , and other applications. Another insightful view put forth is that pure mathematics
3267-451: Is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on problems, often concrete but sometimes abstract. As professionals focused on problem solving, applied mathematicians look into the formulation, study, and use of mathematical models in science , engineering , business , and other areas of mathematical practice. Pure mathematics
3388-405: Is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes. Euler's name is associated with a large number of topics . Euler's work averages 800 pages a year from 1725 to 1783. He also wrote over 4500 letters and hundreds of manuscripts. It has been estimated that Leonhard Euler was the author of
3509-429: Is credited with introducing the modern concept of a function, as opposed to the older vague understanding of a function as an analytic formula. Imre Lakatos cites Hermann Hankel as the early origin of this attribution, but disputes the claim saying that "there is ample evidence that he had no idea of this concept [...] for instance when he discusses piecewise continuous functions, he says that at points of discontinuity,
3630-400: Is not necessarily applied mathematics : it is possible to study abstract entities with respect to their intrinsic nature, and not be concerned with how they manifest in the real world. Even though the pure and applied viewpoints are distinct philosophical positions, in practice there is much overlap in the activity of pure and applied mathematicians. To develop accurate models for describing
3751-657: Is preserved in the department of physiology at the University of Göttingen, along with the brain of Gauss. The Academy in Berlin honored him with a formal memorial speech presented by Kummer in 1860, and later ordered the publication of his collected works edited by Kronecker and Lazarus Fuchs . Number theory was Dirichlet's main research interest, a field in which he found several deep results and in proving them introduced some fundamental tools, many of which were later named after him. In 1837, Dirichlet proved his theorem on arithmetic progressions using concepts from mathematical analysis to tackle an algebraic problem, thus creating
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3872-631: Is the master of us all." Carl Friedrich Gauss wrote: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." His 866 publications and his correspondence are being collected in the Opera Omnia Leonhard Euler which, when completed, will consist of 81 quartos . He spent most of his adult life in Saint Petersburg , Russia, and in Berlin , then
3993-430: Is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not possible: there is no Eulerian circuit . This solution is considered to be the first theorem of graph theory . Euler also discovered the formula V − E + F = 2 {\displaystyle V-E+F=2} relating the number of vertices, edges, and faces of
4114-758: Is well known in analysis for his frequent use and development of power series , the expression of functions as sums of infinitely many terms, such as e x = ∑ n = 0 ∞ x n n ! = lim n → ∞ ( 1 0 ! + x 1 ! + x 2 2 ! + ⋯ + x n n ! ) . {\displaystyle e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}=\lim _{n\to \infty }\left({\frac {1}{0!}}+{\frac {x}{1!}}+{\frac {x^{2}}{2!}}+\cdots +{\frac {x^{n}}{n!}}\right).} Euler's use of power series enabled him to solve
4235-565: The Institutiones calculi differentialis was published. In 1755, he was elected a foreign member of the Royal Swedish Academy of Sciences and of the French Academy of Sciences . Notable students of Euler in Berlin included Stepan Rumovsky , later considered as the first Russian astronomer. In 1748 he declined an offer from the University of Basel to succeed the recently deceased Johann Bernoulli. In 1753 he bought
4356-541: The 1848 revolution ; he even guarded with a rifle the palace of the Prince of Prussia. After the revolution failed, the Military Academy closed temporarily, causing him a large loss of income. When it reopened, the environment became more hostile to him, as officers he was teaching were expected to be loyal to the constituted government. Some of the press who had not sided with the revolution pointed him out, as well as Jacobi and other liberal professors, as "the red contingent of
4477-1040: The Basel problem , finding the sum of the reciprocals of squares of every natural number, in 1735 (he provided a more elaborate argument in 1741). The Basel problem was originally posed by Pietro Mengoli in 1644, and by the 1730s was a famous open problem, popularized by Jacob Bernoulli and unsuccessfully attacked by many of the leading mathematicians of the time. Euler found that: ∑ n = 1 ∞ 1 n 2 = lim n → ∞ ( 1 1 2 + 1 2 2 + 1 3 2 + ⋯ + 1 n 2 ) = π 2 6 . {\displaystyle \sum _{n=1}^{\infty }{1 \over n^{2}}=\lim _{n\to \infty }\left({\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots +{\frac {1}{n^{2}}}\right)={\frac {\pi ^{2}}{6}}.} Euler introduced
4598-459: The Bernoulli numbers , Fourier series , Euler numbers , the constants e and π , continued fractions, and integrals. He integrated Leibniz 's differential calculus with Newton's Method of Fluxions , and developed tools that made it easier to apply calculus to physical problems. He made great strides in improving the numerical approximation of integrals, inventing what are now known as
4719-764: The Gymnasium Bonn [ de ] under the care of Peter Joseph Elvenich , a student his family knew. In 1820, Dirichlet moved to the Jesuit Gymnasium in Cologne , where his lessons with Georg Ohm helped widen his knowledge in mathematics. He left the gymnasium a year later with only a certificate, as his inability to speak fluent Latin prevented him from earning the Abitur . Dirichlet again persuaded his parents to provide further financial support for his studies in mathematics, against their wish for
4840-550: The Imperial Russian Academy of Sciences in Saint Petersburg in 1725, leaving Euler with the assurance they would recommend him to a post when one was available. On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia. When Daniel assumed his brother's position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726, Euler eagerly accepted
4961-676: The Schock Prize , and the Nevanlinna Prize . The American Mathematical Society , Association for Women in Mathematics , and other mathematical societies offer several prizes aimed at increasing the representation of women and minorities in the future of mathematics. Several well known mathematicians have written autobiographies in part to explain to a general audience what it is about mathematics that has made them want to devote their lives to its study. These provide some of
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5082-519: The St. Petersburg Academy , which had retained him as a member and paid him an annual stipend. Euler's Introductio in Analysin Infinitorum was published in two parts in 1748. In addition to his own research, Euler supervised the library, the observatory, the botanical garden, and the publication of calendars and maps from which the academy derived income. He was even involved in the design of
5203-480: The University of Göttingen decided to call Dirichlet as his successor. Given the difficulties faced in Berlin, he decided to accept the offer and immediately moved to Göttingen with his family. Kummer was called to assume his position as a professor of mathematics in Berlin. Dirichlet enjoyed his time in Göttingen, as the lighter teaching load allowed him more time for research and he came into close contact with
5324-490: The Werke , is available in: Mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems . Mathematicians are concerned with numbers , data , quantity , structure , space , models , and change . One of the earliest known mathematicians was Thales of Miletus ( c. 624 – c. 546 BC ); he has been hailed as
5445-642: The central limit theorem . The Dirichlet distribution and the Dirichlet process , based on the Dirichlet integral , are named after him. Dirichlet was elected as a member of several academies: In 1855 Dirichlet was awarded the civil class medal of the Pour le Mérite order at Alexander von Humboldt 's recommendation. The Dirichlet crater on the Moon and the 11665 Dirichlet asteroid are named after him. A complete bibliography of Dirichlet's published works, including translations thereof and lectures not contained in
5566-478: The graduate level . In some universities, a qualifying exam serves to test both the breadth and depth of a student's understanding of mathematics; the students who pass are permitted to work on a doctoral dissertation . Mathematicians involved with solving problems with applications in real life are called applied mathematicians . Applied mathematicians are mathematical scientists who, with their specialized knowledge and professional methodology, approach many of
5687-470: The imaginary unit − 1 {\displaystyle {\sqrt {-1}}} , the Greek letter Σ {\displaystyle \Sigma } (capital sigma ) to express summations , the Greek letter Δ {\displaystyle \Delta } (capital delta ) for finite differences , and lowercase letters to represent the sides of a triangle while representing
5808-527: The largest known prime until 1867. Euler also contributed major developments to the theory of partitions of an integer . In 1735, Euler presented a solution to the problem known as the Seven Bridges of Königsberg . The city of Königsberg , Prussia was set on the Pregel River, and included two large islands that were connected to each other and the mainland by seven bridges. The problem
5929-550: The Berlin Academy and, in 1842, obtained for him a full professor position at the University of Breslau. In 1840 Kummer married Ottilie Mendelssohn, a cousin of Rebecka's. In 1843, when Jacobi fell ill, Dirichlet traveled to Königsberg to help him, then obtained for him the assistance of King Friedrich Wilhelm IV 's personal physician. When the physician recommended that Jacobi spend some time in Italy, Dirichlet joined him on
6050-540: The Euler–Mascheroni constant, and studied its relationship with the harmonic series , the gamma function , and values of the Riemann zeta function . Euler introduced the use of the exponential function and logarithms in analytic proofs . He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex numbers , thus greatly expanding
6171-586: The Italian and German universities, but as they already enjoyed substantial freedoms and autonomy the changes there had begun with the Age of Enlightenment , the same influences that inspired Humboldt. The Universities of Oxford and Cambridge emphasized the importance of research , arguably more authentically implementing Humboldt's idea of a university than even German universities, which were subject to state authority. Overall, science (including mathematics) became
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#17327662317966292-616: The Military School. In 1832 Dirichlet became a member of the Prussian Academy of Sciences , the youngest member at only 27 years old. Dirichlet had a good reputation with students for the clarity of his explanations and enjoyed teaching, especially as his University lectures tended to be on the more advanced topics in which he was doing research: number theory (he was the first German professor to give lectures on number theory), analysis and mathematical physics . He advised
6413-434: The University of Breslau. The probation was extended for three years until the position becoming definite in 1831. After Dirichlet's move to Berlin, Humboldt introduced him to the great salons held by the banker Abraham Mendelssohn Bartholdy and his family. Their house was a weekly gathering point for Berlin artists and scientists, including Abraham's children Felix and Fanny Mendelssohn , both outstanding musicians, and
6534-403: The academy beginning in 1720) for the first time. The problem posed that year was to find the best way to place the masts on a ship. Pierre Bouguer , who became known as "the father of naval architecture", won and Euler took second place. Over the years, Euler entered this competition 15 times, winning 12 of them. Johann Bernoulli's two sons, Daniel and Nicolaus , entered into service at
6655-539: The age of eight, Euler was sent to live at his maternal grandmother's house and enrolled in the Latin school in Basel. In addition, he received private tutoring from Johannes Burckhardt, a young theologian with a keen interest in mathematics. In 1720, at thirteen years of age, Euler enrolled at the University of Basel . Attending university at such a young age was not unusual at the time. The course on elementary mathematics
6776-485: The angles as capital letters. He gave the current definition of the constant e {\displaystyle e} , the base of the natural logarithm , now known as Euler's number . Euler is also credited with being the first to develop graph theory (partly as a solution for the problem of the Seven Bridges of Königsberg , which is also considered the first practical application of topology). He also became famous for, among many other accomplishments, providing
6897-543: The ascension of the twelve-year-old Peter II . The nobility, suspicious of the academy's foreign scientists, cut funding for Euler and his colleagues and prevented the entrance of foreign and non-aristocratic students into the Gymnasium and universities. Conditions improved slightly after the death of Peter II in 1730 and the German-influenced Anna of Russia assumed power. Euler swiftly rose through
7018-464: The best glimpses into what it means to be a mathematician. The following list contains some works that are not autobiographies, but rather essays on mathematics and mathematicians with strong autobiographical elements. Leonhard Euler Leonhard Euler ( / ˈ ɔɪ l ər / OY -lər ; German: [ˈleːɔnhaʁt ˈʔɔʏlɐ] , Swiss Standard German: [ˈleɔnhard ˈɔʏlər] ; 15 April 1707 – 18 September 1783)
7139-500: The branch of analytic number theory . In proving the theorem, he introduced the Dirichlet characters and L-functions . In that article, he also noted the difference between the absolute and conditional convergence of series and its impact in what was later called the Riemann series theorem . In 1841, he generalized his arithmetic progressions theorem from integers to the ring of Gaussian integers Z [ i ] {\displaystyle \mathbb {Z} [i]} . In
7260-417: The capital of Prussia . Euler is credited for popularizing the Greek letter π {\displaystyle \pi } (lowercase pi ) to denote the ratio of a circle's circumference to its diameter , as well as first using the notation f ( x ) {\displaystyle f(x)} for the value of a function, the letter i {\displaystyle i} to express
7381-708: The case n = 5 at the French Academy of Sciences , an exceptional feat for a 20-year-old student with no degree. His lecture at the Academy had also put Dirichlet in close contact with Fourier and Poisson , who raised his interest in theoretical physics , especially Fourier's analytic theory of heat . As General Foy died in November 1825 and he could not find any paying position in France, Dirichlet had to return to Prussia. Fourier and Poisson introduced him to Alexander von Humboldt , who had been called to join
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#17327662317967502-498: The constant γ = lim n → ∞ ( 1 + 1 2 + 1 3 + 1 4 + ⋯ + 1 n − ln ( n ) ) ≈ 0.5772 , {\displaystyle \gamma =\lim _{n\rightarrow \infty }\left(1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+\cdots +{\frac {1}{n}}-\ln(n)\right)\approx 0.5772,} now known as Euler's constant or
7623-576: The continuing turmoil in Russia, Euler left St. Petersburg in June 1741 to take up a post at the Berlin Academy , which he had been offered by Frederick the Great of Prussia . He lived for 25 years in Berlin , where he wrote several hundred articles. In 1748 his text on functions called the Introductio in analysin infinitorum was published and in 1755 a text on differential calculus called
7744-503: The court of King Friedrich Wilhelm III . Humboldt, planning to make Berlin a center of science and research, immediately offered his help to Dirichlet, sending letters in his favour to the Prussian government and to the Prussian Academy of Sciences . Humboldt also secured a recommendation letter from Gauss, who upon reading his memoir on Fermat's theorem wrote with an unusual amount of praise that "Dirichlet showed excellent talent". With
7865-474: The curriculum, raising the level of scientific education there. However, he gradually started feeling that his double teaching load, at the Military academy and at the university, was limiting the time available for his research. While in Berlin, Dirichlet kept in contact with other mathematicians. In 1829, during a trip, he met Carl Jacobi , at the time professor of mathematics at Königsberg University . Over
7986-422: The damage caused to Euler's estate, with Empress Elizabeth of Russia later adding a further payment of 4000 rubles—an exorbitant amount at the time. Euler decided to leave Berlin in 1766 and return to Russia. During his Berlin years (1741–1766), Euler was at the peak of his productivity. He wrote 380 works, 275 of which were published. This included 125 memoirs in the Berlin Academy and over 100 memoirs sent to
8107-537: The development of the prime number theorem . Euler's interest in number theory can be traced to the influence of Christian Goldbach , his friend in the St. Petersburg Academy. Much of Euler's early work on number theory was based on the work of Pierre de Fermat . Euler developed some of Fermat's ideas and disproved some of his conjectures, such as his conjecture that all numbers of the form 2 2 n + 1 {\textstyle 2^{2^{n}}+1} ( Fermat numbers ) are prime. Euler linked
8228-520: The doctoral theses of several important German mathematicians, as Gotthold Eisenstein , Leopold Kronecker , Rudolf Lipschitz and Carl Wilhelm Borchardt , while being influential in the mathematical formation of many other scientists, including Elwin Bruno Christoffel , Wilhelm Weber , Eduard Heine , Ludwig von Seidel and Julius Weingarten . At the Military Academy, Dirichlet managed to introduce differential and integral calculus in
8349-428: The entirety of the poem, along with stating the first and last sentence on each page of the edition from which he had learnt it. Euler's eyesight worsened throughout his mathematical career. In 1738, three years after nearly expiring from fever, he became almost blind in his right eye. Euler blamed the cartography he performed for the St. Petersburg Academy for his condition, but the cause of his blindness remains
8470-523: The first boundary-value problem , for the Laplace equation , proving the uniqueness of the solution; this type of problem in the theory of partial differential equations was later named the Dirichlet problem after him. A function satisfying a partial differential equation subject to the Dirichlet boundary conditions must have fixed values on the boundary. In the proof he notably used the principle that
8591-435: The first advance in the theorem since Fermat 's own proof of the case n = 4 and Euler 's proof for n = 3 . Adrien-Marie Legendre , one of the referees, soon completed the proof for this case; Dirichlet completed his own proof a short time after Legendre, and a few years later produced a full proof for the case n = 14 . In June 1825 he was accepted to lecture on his partial proof for
8712-572: The first results by introducing the Dirichlet hyperbola method , is still an unsolved problem in number theory despite later contributions by other mathematicians. Inspired by the work of his mentor in Paris, Dirichlet published in 1829 a famous memoir giving the conditions , showing for which functions the convergence of the Fourier series holds. Before Dirichlet's solution, not only Fourier, but also Poisson and Cauchy had tried unsuccessfully to find
8833-476: The first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry , by deriving four corollaries to Thales's theorem . The number of known mathematicians grew when Pythagoras of Samos ( c. 582 – c. 507 BC ) established the Pythagorean school , whose doctrine it
8954-442: The focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. The German university system fostered professional, bureaucratically regulated scientific research performed in well-equipped laboratories, instead of
9075-1060: The following. There is no Nobel Prize in mathematics, though sometimes mathematicians have won the Nobel Prize in a different field, such as economics or physics. Prominent prizes in mathematics include the Abel Prize , the Chern Medal , the Fields Medal , the Gauss Prize , the Nemmers Prize , the Balzan Prize , the Crafoord Prize , the Shaw Prize , the Steele Prize , the Wolf Prize ,
9196-668: The force of the wheels necessary to raise the water to a reservoir, from where it should fall back through channels, finally spurting out in Sanssouci . My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces to the reservoir. Vanity of vanities! Vanity of geometry! However, the disappointment was almost surely unwarranted from a technical perspective. Euler's calculations look likely to be correct, even if Euler's interactions with Frederick and those constructing his fountain may have been dysfunctional. Throughout his stay in Berlin, Euler maintained
9317-625: The function has two values". Dirichlet also worked in mathematical physics , lecturing and publishing research in potential theory (including the Dirichlet problem and Dirichlet principle mentioned above), the theory of heat and hydrodynamics . He improved on Lagrange 's work on conservative systems by showing that the condition for equilibrium is that the potential energy is minimal. Dirichlet also lectured on probability theory and least squares , introducing some original methods and results, in particular for limit theorems and an improvement of Laplace's method of approximation related to
9438-633: The imposing problems presented in related scientific fields. With professional focus on a wide variety of problems, theoretical systems, and localized constructs, applied mathematicians work regularly in the study and formulation of mathematical models . Mathematicians and applied mathematicians are considered to be two of the STEM (science, technology, engineering, and mathematics) careers. The discipline of applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry; thus, "applied mathematics"
9559-414: The integer n that are coprime to n . Using properties of this function, he generalized Fermat's little theorem to what is now known as Euler's theorem . He contributed significantly to the theory of perfect numbers , which had fascinated mathematicians since Euclid . He proved that the relationship shown between even perfect numbers and Mersenne primes (which he had earlier proved) was one-to-one,
9680-580: The kind of research done by private and individual scholars in Great Britain and France. In fact, Rüegg asserts that the German system is responsible for the development of the modern research university because it focused on the idea of "freedom of scientific research, teaching and study." Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education , and then proceed to specialize in topics of their own choice at
9801-421: The king of Prussia , Fredrick William III , to build a university in Berlin based on Friedrich Schleiermacher 's liberal ideas; the goal was to demonstrate the process of the discovery of knowledge and to teach students to "take account of fundamental laws of science in all their thinking." Thus, seminars and laboratories started to evolve. British universities of this period adopted some approaches familiar to
9922-463: The left eye as well. However, his condition appeared to have little effect on his productivity. With the aid of his scribes, Euler's productivity in many areas of study increased; and, in 1775, he produced, on average, one mathematical paper every week. In St. Petersburg on 18 September 1783, after a lunch with his family, Euler was discussing the newly discovered planet Uranus and its orbit with Anders Johan Lexell when he collapsed and died from
10043-483: The modern notation for the trigonometric functions , the letter e for the base of the natural logarithm (now also known as Euler's number ), the Greek letter Σ for summations and the letter i to denote the imaginary unit . The use of the Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it originated with Welsh mathematician William Jones . The development of infinitesimal calculus
10164-502: The motion of rigid bodies . He also made substantial contributions to the study of elastic deformations of solid objects. Leonhard Euler was born on 15 April 1707, in Basel to Paul III Euler, a pastor of the Reformed Church , and Marguerite (née Brucker), whose ancestors include a number of well-known scholars in the classics. He was the oldest of four children, having two younger sisters, Anna Maria and Maria Magdalena, and
10285-615: The names Rebecka Henriette Mendelssohn Bartholdy. She became a part of the notable salon of her parents, Abraham Mendelssohn and his wife Lea, having social contacts with the important musicians, artists and scientists in a highly creative period of German intellectual life. In 1829 she sang a small role in the premiere, given at the Mendelssohn house, of Felix's Singspiel Die Heimkehr aus der Fremde . She later wrote: My older brother and sister stole my reputation as an artist. In any other family I would have been highly regarded as
10406-455: The nature of prime distribution with ideas in analysis. He proved that the sum of the reciprocals of the primes diverges . In doing so, he discovered the connection between the Riemann zeta function and prime numbers; this is known as the Euler product formula for the Riemann zeta function . Euler invented the totient function φ( n ), the number of positive integers less than or equal to
10527-454: The new generation of researchers, especially Richard Dedekind and Bernhard Riemann . After moving to Göttingen he was able to obtain a small annual stipend for Riemann to retain him in the teaching staff there. Dedekind, Riemann, Moritz Cantor and Alfred Enneper , although they had all already earned their PhDs, attended Dirichlet's classes to study with him. Dedekind, who felt that there were gaps in his mathematics education, considered that
10648-541: The notion of a mathematical function . He is also known for his work in mechanics , fluid dynamics , optics , astronomy , and music theory . Euler is regarded as one of the greatest, most prolific mathematicians in history and the greatest of the 18th century. Several great mathematicians who produced their work after Euler's death have recognised his importance in the field as shown by quotes attributed to many of them: Pierre-Simon Laplace expressed Euler's influence on mathematics by stating, "Read Euler, read Euler, he
10769-435: The occasion to study with Dirichlet made him "a new human being". He later edited and published Dirichlet's lectures and other results in number theory under the title Vorlesungen über Zahlentheorie ( Lectures on Number Theory ). In the summer of 1858, during a trip to Montreux , Dirichlet suffered a heart attack. On 5 May 1859, he died in Göttingen, several months after the death of his wife Rebecka. Dirichlet's brain
10890-560: The offer, but delayed making the trip to Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg in May 1727. He was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian, settled into life in Saint Petersburg and took on an additional job as
11011-571: The painter Wilhelm Hensel (Fanny's husband). Dirichlet showed great interest in Abraham's daughter Rebecka, whom he married in 1832. Rebecka Henriette Lejeune Dirichlet (née Rebecka Mendelssohn; 11 April 1811 – 1 December 1858) was a granddaughter of Moses Mendelssohn and the youngest sister of Felix Mendelssohn and Fanny Mendelssohn . Rebecka was born in Hamburg . In 1816 her parents arranged for her to be baptised at which point she took
11132-403: The polar opposite of Voltaire , who enjoyed a high place of prestige at Frederick's court. Euler was not a skilled debater and often made it a point to argue subjects that he knew little about, making him the frequent target of Voltaire's wit. Frederick also expressed disappointment with Euler's practical engineering abilities, stating: I wanted to have a water jet in my garden: Euler calculated
11253-531: The probability and likely cost of the occurrence of an event such as death, sickness, injury, disability, or loss of property. Actuaries also address financial questions, including those involving the level of pension contributions required to produce a certain retirement income and the way in which a company should invest resources to maximize its return on investments in light of potential risk. Using their broad knowledge, actuaries help design and price insurance policies, pension plans, and other financial strategies in
11374-728: The problem by awarding him an honorary doctorate in February 1827. Also, the Minister of Education granted him a dispensation for the Latin disputation required for the Habilitation . Dirichlet earned the Habilitation and lectured in the 1827–28 year as a Privatdozent at Breslau . While in Breslau, Dirichlet continued his number-theoretic research, publishing important contributions to the biquadratic reciprocity law which at
11495-520: The ranks in the academy and was made a professor of physics in 1731. He also left the Russian Navy, refusing a promotion to lieutenant . Two years later, Daniel Bernoulli, fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department. In January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell . Frederick II had made an attempt to recruit
11616-484: The real world, many applied mathematicians draw on tools and techniques that are often considered to be "pure" mathematics. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research. Many professional mathematicians also engage in the teaching of mathematics. Duties may include: Many careers in mathematics outside of universities involve consulting. For instance, actuaries assemble and analyze data to estimate
11737-504: The scope of mathematical applications of logarithms. He also defined the exponential function for complex numbers and discovered its relation to the trigonometric functions . For any real number φ (taken to be radians), Euler's formula states that the complex exponential function satisfies e i φ = cos φ + i sin φ {\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi } which
11858-622: The services of Euler for his newly established Berlin Academy in 1740, but Euler initially preferred to stay in St Petersburg. But after Empress Anna died and Frederick II agreed to pay 1600 ecus (the same as Euler earned in Russia) he agreed to move to Berlin. In 1741, he requested permission to leave to Berlin, arguing he was in need of a milder climate for his eyesight. The Russian academy gave its consent and would pay him 200 rubles per year as one of its active members. Concerned about
11979-403: The seventeenth century at Oxford with the scientists Robert Hooke and Robert Boyle , and at Cambridge where Isaac Newton was Lucasian Professor of Mathematics & Physics . Moving into the 19th century, the objective of universities all across Europe evolved from teaching the "regurgitation of knowledge" to "encourag[ing] productive thinking." In 1810, Alexander von Humboldt convinced
12100-505: The solution is the function that minimizes the so-called Dirichlet energy . Riemann later named this approach the Dirichlet principle , although he knew it had also been used by Gauss and by Lord Kelvin . While trying to gauge the range of functions for which convergence of the Fourier series can be shown, Dirichlet defines a function by the property that "to any x there corresponds a single finite y ", but then restricts his attention to piecewise continuous functions. Based on this, he
12221-421: The staff". In 1849 Dirichlet participated, together with his friend Jacobi, in the jubilee of Gauss's doctorate. Despite Dirichlet's expertise and the honours he received, and even though, by 1851, he had finally completed all formal requirements for a full professor, the issue of raising his pay at the university still dragged on and he was still unable to leave the Military Academy. In 1855, upon Gauss's death,
12342-480: The subject of speculation. Euler's vision in that eye worsened throughout his stay in Germany, to the extent that Frederick referred to him as " Cyclops ". Euler remarked on his loss of vision, stating "Now I will have fewer distractions." In 1766 a cataract in his left eye was discovered. Though couching of the cataract temporarily improved his vision, complications ultimately rendered him almost totally blind in
12463-469: The support of Humboldt and Gauss, Dirichlet was offered a teaching position at the University of Breslau . However, as he had not passed a doctoral dissertation, he submitted his memoir on the Fermat theorem as a thesis to the University of Bonn . Again his lack of fluency in Latin rendered him unable to hold the required public disputation of his thesis; after much discussion, the university decided to bypass
12584-499: The time was a focal point of Gauss's research. Alexander von Humboldt took advantage of these new results, which had also drawn enthusiastic praise from Friedrich Bessel , to arrange for him the desired transfer to Berlin. Given Dirichlet's young age (he was 23 years old at the time), Humboldt was able to get him only a trial position at the Prussian Military Academy in Berlin while remaining nominally employed by
12705-684: The time was part of the First French Empire , reverting to Prussia after the Congress of Vienna in 1815. His father Johann Arnold Lejeune Dirichlet was the postmaster, merchant, and city councilor. His paternal grandfather had come to Düren from Richelette (or more likely Richelle [ fr ] ), a small community 5 km (3 miles) north east of Liège in Belgium , from which his surname "Lejeune Dirichlet" (" le jeune de Richelette ", French for "the youth from Richelette")
12826-501: The transfer and in 1831 assigned him to the faculty of philosophy . The faculty required him to undertake a renewed habilitation qualification, and although Dirichlet wrote a Habilitationsschrift as needed, he postponed giving the mandatory lecture in Latin for another 20 years, until 1851. As he had not completed this formal requirement, he remained attached to the faculty with less than full rights, including restricted emoluments, forcing him to keep in parallel his teaching position at
12947-407: The trip together with his family. They were accompanied to Italy by Ludwig Schläfli , who came as a translator; as he was strongly interested in mathematics, both Dirichlet and Jacobi lectured to him during the trip, and he later became an important mathematician himself. The Dirichlet family extended their stay in Italy to 1845, their daughter Flora being born there. In 1844, Jacobi moved to Berlin as
13068-465: The water fountains at Sanssouci , the King's summer palace. The political situation in Russia stabilized after Catherine the Great's accession to the throne, so in 1766 Euler accepted an invitation to return to the St. Petersburg Academy. His conditions were quite exorbitant—a 3000 ruble annual salary, a pension for his wife, and the promise of high-ranking appointments for his sons. At the university he
13189-440: The years they kept meeting and corresponding on research matters, in time becoming close friends. In 1839, during a visit to Paris, Dirichlet met Joseph Liouville , the two mathematicians becoming friends, keeping in contact and even visiting each other with the families a few years later. In 1839, Jacobi sent Dirichlet a paper by Ernst Kummer , at the time a schoolteacher. Realizing Kummer's potential, they helped him get elected in
13310-943: Was Al-Khawarizmi . A notable feature of many scholars working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics , maths and astronomy of Ibn al-Haytham . The Renaissance brought an increased emphasis on mathematics and science to Europe. During this period of transition from a mainly feudal and ecclesiastical culture to a predominantly secular one, many notable mathematicians had other occupations: Luca Pacioli (founder of accounting ); Niccolò Fontana Tartaglia (notable engineer and bookkeeper); Gerolamo Cardano (earliest founder of probability and binomial expansion); Robert Recorde (physician) and François Viète (lawyer). As time passed, many mathematicians gravitated towards universities. An emphasis on free thinking and experimentation had begun in Britain's oldest universities beginning in
13431-502: Was Johann Albrecht Euler , whose godfather was Christian Goldbach . Three years after his wife's death in 1773, Euler married her half-sister, Salome Abigail Gsell (1723–1794). This marriage lasted until his death in 1783. His brother Johann Heinrich settled in St. Petersburg in 1735 and was employed as a painter at the academy. Early in his life, Euler memorized the entirety of the Aeneid by Virgil , and by old age, could recite
13552-405: Was a Swiss mathematician , physicist , astronomer , geographer , logician , and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory , complex analysis , and infinitesimal calculus . He introduced much of modern mathematical terminology and notation , including
13673-553: Was assisted by his student Anders Johan Lexell . While living in St. Petersburg, a fire in 1771 destroyed his home. On 7 January 1734, he married Katharina Gsell (1707–1773), daughter of Georg Gsell , a painter from the Academy Gymnasium in Saint Petersburg. The young couple bought a house by the Neva River . Of their thirteen children, only five survived childhood, three sons and two daughters. Their first son
13794-525: Was at the forefront of 18th-century mathematical research, and the Bernoullis —family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of mathematical rigour (in particular his reliance on the principle of the generality of algebra ), his ideas led to many great advances. Euler
13915-803: Was buried next to Katharina at the Smolensk Lutheran Cemetery on Vasilievsky Island . In 1837, the Russian Academy of Sciences installed a new monument, replacing his overgrown grave plaque. To commemorate the 250th anniversary of Euler's birth in 1957, his tomb was moved to the Lazarevskoe Cemetery at the Alexander Nevsky Monastery . Euler worked in almost all areas of mathematics, including geometry , infinitesimal calculus , trigonometry , algebra , and number theory , as well as continuum physics , lunar theory , and other areas of physics . He
14036-427: Was called "the most remarkable formula in mathematics" by Richard Feynman . A special case of the above formula is known as Euler's identity , e i π + 1 = 0 {\displaystyle e^{i\pi }+1=0} Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations . He found
14157-408: Was derived. Although his family was not wealthy and he was the youngest of seven children, his parents supported his education. They enrolled him in an elementary school and then private school in hope that he would later become a merchant. The young Dirichlet, who showed a strong interest in mathematics before age 12, persuaded his parents to allow him to continue his studies. In 1817 they sent him to
14278-423: Was given by Johann Bernoulli , the younger brother of the deceased Jacob Bernoulli (who had taught Euler's father). Johann Bernoulli and Euler soon got to know each other better. Euler described Bernoulli in his autobiography: It was during this time that Euler, backed by Bernoulli, obtained his father's consent to become a mathematician instead of a pastor. In 1723, Euler received a Master of Philosophy with
14399-431: Was ongoing throughout the reign of certain caliphs, and it turned out that certain scholars became experts in the works they translated, and in turn received further support for continuing to develop certain sciences. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences. An example of a translator and mathematician who benefited from this type of support
14520-550: Was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypatia of Alexandria ( c. AD 350 – 415). She succeeded her father as librarian at the Great Library and wrote many works on applied mathematics. Because of
14641-544: Was translated into multiple languages, published across Europe and in the United States, and became more widely read than any of his mathematical works. The popularity of the Letters testifies to Euler's ability to communicate scientific matters effectively to a lay audience, a rare ability for a dedicated research scientist. Despite Euler's immense contribution to the academy's prestige and having been put forward as
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