Mathematical international phonetic alphabet

Leonhard Euler

Leonhard euler (pronounced OilerIPA [? l？ ) (April 15,1707–September18 [O.S. September 7] 1783) is a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. He has published more papers than any mathematician in history. [ 1]

Euler made important discoveries in different fields such as calculus and topology. He also introduced many modern mathematical terms and symbols, especially for mathematical analysis, such as the concept of mathematical function. [2] He is also famous for his work in mechanics, optics and astronomy.

Euler is considered as the most outstanding mathematician in the18th century and one of the greatest mathematicians in history. He is also one of the most prolific writers; His collection of works includes 60-80 quartets. [3] Pierre-Simon Laplace's words expressed Euler's influence on mathematics: "Read Euler, read Euler, he is the master of all of us". [4]

Euler appeared on the sixth series of Swiss 10 franc banknotes [5] and many stamps from Switzerland, Germany and Russia. The asteroid 2002 Euler was named after him. The Lutheran Church also commemorated him in the Saint's Calendar on May 24th.

Content [hide]

1 biography

1. 1 childhood

1.2 St. Petersburg

1.3 Berlin

1.4 vision degradation

1.5 the last stage of life

2 Contribution to Mathematics

2. 1 mathematical symbol

2.2 analysis

2.3 number theory

2.4 Graph Theory

2.5 Applied Mathematics

2.6 Physics and Astronomy

2.7 Logic

3 Philosophy and religious beliefs

4 Bibliography

5 See also

6 comments

7 further reading

8 external links

[editor] biography

[Editor] Childhood

Swiss 10 franc notes in memory of the most successful Swiss mathematician Euler in history. Euler was born in Basel to Paul Euler, a pastor of the Reformed Church, and Margaret Brooke, the pastor's daughter. He has two sisters, Anna Maria and Maria Magdalena. Shortly after Leonhard was born, Euler's family moved from Basel to Liheng town, where Euler spent most of his childhood. Paul Euler was a friend of the Bernoulli family, and johann bernoulli, considered as the most important mathematician in Europe at that time, finally had an important influence on young Leonhard. His early formal education began in Basel, where he was sent to live with his grandmother. /kloc-at the age of 0/3, he was admitted to university of basel, and in 1723, he obtained a master's degree in philosophy with a paper comparing Descartes and Newton's philosophy. At this time, he was taking a Saturday afternoon class in johann bernoulli, and he soon discovered his new student's incredible talent in mathematics. [6]

At this time, Euler was urged by his father to study theology, Greek and Hebrew in order to become a priest. Johann bernoulli intervened and convinced Paul Euler that Leonhard was destined to be a great mathematician. 1726, Euler completed his doctoral thesis on sound propagation, entitled De Sono [7], 1727, and he participated in the Paris Academy Award Competition. The problem of that year was to find the best way to place the mast on the ship. He won the second place, only lost to pierre bouguer, a man who is now called "the father of shipbuilding". However, Euler finally won 12 coveted annual awards in his career. [8]

[editor] St. Petersburg

Around this time, johann bernoulli's two sons, Daniel and Nicholas, worked in the Royal Russian Academy of Sciences in St Petersburg. 1726 In July, Nicholas died of appendicitis after staying in Russia for one year. When Daniel took over his brother's position in the math/physics department, he recommended that his vacant physiological position be filled by his friend Euler. 1 1 month, Euler eagerly accepted the proposal, but delayed his trip to St. Petersburg. During this period, he applied for the position of physics professor in university of basel, but failed. [9]

1957 stamps commemorating Euler's 250th birthday in the former Soviet Union. Text: It is 250 years since the birth of the great mathematician Academician An. Euler arrived in the Russian capital on May 17 and 1727. He was promoted from a junior position in the medical department of the college to the department of mathematics. He stayed at daniel bernoulli's house and often worked closely with him. Euler mastered Russian and settled in St. Petersburg. He also took an extra job as a military doctor in the Russian navy. [ 10]

St Petersburg College, established by Peter the Great, aims to improve the education level of Russia and narrow the scientific gap with Western Europe. Therefore, it is particularly attractive to foreign scholars like Euler: the college has sufficient financial resources and a comprehensive library extracted from Peter's own and aristocratic private libraries. Only a few students register in the college to reduce the teaching burden of teachers. The college emphasizes research and provides teachers with time and freedom to study scientific problems. [8]

However, Catherine I, the patron of the college, tried to continue her late husband's progressive policy and died of Euler's arrival. After twelve-year-old Peter Alekseyevich Romanov ascended the throne, Russian aristocrats gained power. The nobles were suspicious of foreign scientists in the Academy of Sciences, so they cut their funds, which caused many other difficulties for Euler and his colleagues.

After Peter Alekseyevich Romanov's death, the situation improved slightly. Euler was promoted rapidly in the college and became a professor of physics in 173 1. Two years later, daniel bernoulli was fed up with the censorship and hostility he faced in St. Petersburg and left Basel. Euler succeeded him as the head of the Department of Mathematics. [ 1 1]

1734 1 7, he married Katarina Gesell, the daughter of a painter in the college gymnasium. The young couple bought a house by the Neva River and gave birth to thirteen children, only five of whom lived through childhood. [ 12]

[Editor] Berlin

Stamps of the former German Democratic Republic commemorating the 200th anniversary of Euler's death. In the middle is to show his polyhedron formula. Out of concern about the continued turmoil in Russia, Euler debated whether to stay in St. Petersburg. Frederick the great of Prussia offered him a position at the Berlin Academy, and he accepted it. He left St. Petersburg on June 9 and lived in Berlin for 25 years, where he wrote more than 380 articles. In Berlin, he published two most famous works: Introduction to Infinite Analysis, a book on functions published in 1748, and Differential System, a book on differential calculus. [ 13]

In addition, Euler was asked to tutor Princess Anhalt-Dessau, Frederick's niece. He wrote more than 200 letters to her, which were later compiled into bestsellers, entitled "Letters from Euler to a German princess on different themes of natural philosophy". This book contains Euler's exposition of various topics related to physics and mathematics, and provides valuable insights into Euler's personality and religious beliefs. This book was eventually read more widely than any of his mathematical works, and was published all over Europe and the United States. The popularity of these letters proves Euler's ability to effectively convey scientific materials to laymen, which is rare for a scientist who focuses on research. [ 13]

Although Euler made great contributions to the reputation of the college, he was forced to leave Berlin in the end. This is partly due to the personality conflict with Frederick. Frederick began to think that he was not clever, especially compared with the circle of philosophers brought to the college by the German king. Voltaire was one of Frederick's employees, and the Frenchman enjoyed a superior position in the king's social circle. Euler is a devout believer and a diligent worker. His beliefs and tastes are very traditional. He is the opposite of Voltaire in many ways. Euler's training in rhetoric is very limited, and he tends to debate things he knows little, which makes him the object of Voltaire's frequent jokes. [13] Frederick also expressed disappointment with Euler's practical engineering ability:

I want to have a sprinkler in my garden: Euler calculated the force of the wheel needed to lift the water to a reservoir, from which the water should flow back through the channel and finally spray out in Sansucci. My mill is geometric. I can't lift a mouthful of water less than fifty paces from the reservoir. Vanity. Vanity! The void of geometry! [ 14]

[editor] vision degradation

Portrait of Emmanuel Handmann 1753. This description implies a problem with the right eyelid, and Euler may suffer from strabismus. The left eye looks healthy because it was destroyed by the later cataract. During Euler's mathematics career, his eyesight became worse and worse. Three years after 1735 suffered from an almost fatal fever, his right eye was almost blind, but Euler attributed his condition to his hard drawing work for St. Petersburg College. The scene in Euler's eyes changed constantly during his stay in Germany, so that Frederick called him "Cyclops". Euler later suffered from cataract in his left eye, and a few weeks after his discovery, he was almost completely blind. Even so, his situation seems to have no effect on his productivity, because he makes up for it with his mental arithmetic skills and photographic memory. For example, Euler can recite Virgil's Aeneas from beginning to end without hesitation, and he can point out the first and last lines of each page. [3]

[Editor] The last stage of life

Alexander nevsky Laura's Tomb of Euler. Since Catherine II ascended the throne, the situation in Russia has greatly improved. 1766, Euler accepted the invitation and returned to St. Petersburg College, where he spent the rest of his life. His second stay in this country was ruined by tragedy. A 177 1 fire in St. Petersburg made him lose his home and almost lost his life. 1773, he lost his wife of 40 years. Euler will remarry in three years.

18, 1783 In September, Euler died of cerebral hemorrhage in St. Petersburg and was buried in Lola, alexander nevsky. Marquis Condorcet, a French mathematician and philosopher, wrote a eulogy for the French Academy of Sciences, and Nicolas von Foss, Euler's son-in-law and secretary of the Royal Academy of Sciences in St. Petersburg, wrote an account of his life and a series of works. Condorcet commented,

"... ilcessa de calculator et de vivre," (he stopped calculating and living). [ 16]

[Editor] Contribution to Mathematics

Euler dabbled in almost all fields of mathematics: geometry, calculus, trigonometry, algebra and number theory, not to mention continuum physics, moon theory and other physical fields. His importance in the history of mathematics cannot be overemphasized: if published, his works, many of which are of fundamental significance, will occupy 60 to 80 quartets [3], and Euler's name is associated with quite a few themes. The 20th century Hungarian mathematician Paul Elder? S may be another mathematician who is considered prolific.

[Edit] Mathematical symbols

Euler introduced and popularized several notation methods through a large number of widely circulated textbooks. Most notably, he introduced the concept of function [2] and was the first to write f(x) to represent the function f applied to the independent variable X. He also introduced modern symbols of trigonometric functions. The letter E stands for the base of natural logarithm (now also called Euler number), the Greek letter ∑ stands for summation, and the letter I stands for imaginary unit. The ratio of the circumference to the diameter of a circle expressed by the Greek letter π was also popularized by Euler, although it was not his initiative. [18] Euler also contributed to the development of complex number system (notation system with a+bi defining negative roots). [ 19]

[edit] analysis

The development of calculus is at the forefront of mathematical research in the18th century, and the Bernoulli family, a family friend of Euler, made great contributions to the early development of this field. Because of their influence, learning calculus naturally became the main focus of Euler's work. Although some of Euler's proofs may be unacceptable under strict modern standards, [20] his thoughts have led to many great progress.

He is famous for frequently using and developing power series in his analysis: that is, expressing a function as the sum of infinite terms, such as

Notably, Euler discovered the power series expansion of e and arc tangent function. His bold use of power series (technically incorrect according to modern standards) enabled him to solve the famous Basel problem in 1735:[20].

Geometric interpretation of Euler formula Euler introduced the application of exponential function and logarithm in analytical proof. He discovered the method of expressing various logarithmic functions by power series, and successfully defined the logarithm of negative numbers and complex numbers, thus greatly expanding the application range of logarithm in mathematics. [17] He also defined the exponential function of complex numbers and found its relationship with trigonometric functions. For any real number φ, Euler formula shows that the complex exponential function satisfies

A special case of the above formula is called Euler identity,

It is called "the most outstanding formula in mathematics" by richard feynman because it uses the concepts of addition, multiplication, exponent and equation alone, and uses the important constants 0, 1, e, i and π alone. [2 1]

In addition, Euler expounded the theory of higher-order transcendental function by introducing γ function, and introduced a new method to solve quartic equation. He also discovered the method of calculating the integral with complex limit, which predicted the development of modern complex analysis, and invented the variational method, including its most famous result-Euler-Lagrange equation.

Euler also took the lead in using analytical methods to solve the problem of number theory. In this process, he combined two completely different branches of mathematics and introduced a new research field-analytic number theory. In the process of opening up this new field, Euler founded hypergeometric series, q- series, hyperbolic trigonometric function and continued fraction analysis theory. For example, he used the divergence of harmonic series to prove the infinity of prime numbers, and used analytical methods to have some understanding of the distribution of prime numbers. Euler's work in this field led to the development of prime number theorem. [22]

[editor] number theory

Euler's keen interest in number theory can be traced back to the influence of his friend Christian Goldbach in St. Petersburg College. Many of his early work on number theory was based on Pierre de Fermat's works. Euler developed some of Fermat's ideas, while denying some of his more eccentric conjectures.

One of the key points of Euler's work is to link the properties of prime number distribution with the ideas in analysis. He proved that the sum of reciprocal prime numbers is divergent. In this process, he discovered the connection between Riemannian Zeta function and prime number, that is, the Euler product formula of Riemannian Zeta function.

Euler proved Newton's identity, Fermat's theorem and Fermat's theorem about the sum of two squares, and made outstanding contributions to Lagrange's four squares theorem. He also invented the all-factorial function φ(n), which specifies the number of positive integers less than n for positive integer n and is coprime with n. Using the properties of this function, he can extend Fermat's Theorem to what was later called euler theorem. He also made a great contribution to the understanding of perfect numbers, which have fascinated mathematicians since Euclid. Euler made progress in the prime number theorem and speculated the law of quadratic reciprocity. These two concepts are regarded as the basic theorems of number theory, and his thoughts paved the way for C.F.Gauss. [23]

Graph theory