Karl Theodore William Karl Theodor Wilhelm Weierstra? , whose surname can be written as Weierstrass,181510/0/0/3/012/897 19), a German mathematician, is known as "modern". Hostains Feld (now German) was born in westfalen and died in Berlin.
Karl Veiershtrass's father is a government official Wilhem Weierstrass and his mother is Tiodola von der Forster. He became interested in mathematics when he was studying in middle school, but after graduating from high school, he entered Bonn University to work in a government department. What he wants to study is law, economy and finance, which goes against his desire to study mathematics. His solution to the contradiction is not to pay attention to the assigned courses, but to continue to teach himself mathematics in private. As a result, he left the university without getting a degree. His father found a place for him in a teacher training school in Mü nster, and he later registered as a teacher in the city. During his study, he took Christoph Gudermann's class and became interested in elliptic functions.
After 1850, Veiershtrass was ill for a long time, but he still published a paper, which made him famous. 1857 Berlin University gave him a math chair.
1854 published a monograph on the development of Abelian function theory-On Abelian function theory, which was made public. According to his academic achievements, Konigsberg University awarded him an honorary doctorate. 1856 was recommended by Cuomo as the University of Berlin (Freie Universit? T Berlin) assistant professor, 1865 promoted to professor. Before his death, most of his research results were taught and disseminated to students. 1886 published a collection of essays on function theory. Although his works are not many, he has published the most influential papers.
Wilstrass's main contributions are in mathematical analysis, analytic function theory, variational method, differential geometry and linear algebra. He is a master who introduces strict arguments into analysis. His critical spirit had a great influence on mathematics in19th century. On the basis of strict logic, he established the theory of real numbers, defined irrational numbers with monotone bounded sequences, and gave strict definitions of upper and lower bounds, limit points and continuous functions of number sets. In 186 1, he also constructed a famous continuous function with differentiability everywhere, which made an important contribution to the arithmeticization of analysis. He completed the definition of limit described by inequality introduced by Cauchy (so-called ε-δ definition). In analytic function theory, Wilstrass also made important contributions. He established the power series expansion theorem of analytic function and the basic theory of multivariate analytic function, and made some achievements in algebraic function theory and Abel integral. In the variational method, he gave the variational structure with parametric function and studied the discontinuous solution of the variational problem. In differential geometry, he studied geodesics and minimal surfaces. In sexual algebra, elementary factor theory is established and used to simplify matrices. He is also an outstanding educator, who has trained a large number of outstanding mathematical talents in his life, including Kovalevskaya, Schwartz, Tammy Leffler, Schottky and fuchs.
Juvenile mathematical genius
1September 826 17, Riemann (1826-1866) was born in a village priest's home in Bre Slentz, Hanover, and was the second of six children.
Riemann loved mathematics since he was a child. At the age of six, he began to learn arithmetic, which showed his mathematical genius. He can not only solve all the math problems left to him, but also often ask some questions to play tricks on his brothers and sisters. /kloc-When I was 0/0 years old, I studied advanced arithmetic and geometry with a professional teacher, and soon surpassed the teacher and often gave better answers to some questions.
Riemann 14 years old went to middle school in Hanover. Due to financial constraints, he always walks back and forth between Hanover and small rural villages. Of course, he has no money to buy reference books. Fortunately, the headmaster of the middle school discovered his talent in mathematics in time. Considering his financial difficulties, the headmaster authorized Riemann to borrow math books from his private library. On the recommendation of the headmaster, Riemann borrowed a mathematician Legendre's Number Theory, which is a four-page masterpiece with 859 pages. Riemann cherished this opportunity to study very much, and he taught himself eagerly. Six days later, Li Man finished learning and returned the book. The headmaster asked him, "How much have you read?" Riemann said, "This is an amazing book, and I have mastered it." A few months later, the headmaster tested him on the content of the book. Riemann answers questions like water and answers comprehensively. Using the principal's library, Riemann also took the time to teach himself the works of Euler, a great mathematician, and thus mastered calculus and its branches. Riemann not only learned mathematics knowledge from Euler's works, but also learned Euler's skills in studying mathematics.
College career
19 years old, Riemann entered the University of G? ttingen. In order to help his family financially and find a paid job as soon as possible, he first studied philosophy and theology, but in addition to these two courses, he also took part in mathematics and physics courses. He listened to Stern's lecture on equation theory and definite integral, Gauss's lecture on least square method and Gordes Mitter's lecture on geomagnetism, and he became interested in mathematics.
Riemann told his father all this and asked for permission to change his major in mathematics. Father heartily agreed to his request. Riemann was very happy and deeply grateful to his father.
1847, in order to learn from more masters, Riemann transferred to Berlin University, where he studied under the great mathematicians jacoby, Dirichlet, Steiner and Eisenstein. He learned advanced mechanics and algebra from jacoby, number theory and analysis from Dirichlet, modern geometry from Stayner and elliptic function theory from Weinstein.
During this period, he was so diligent that he didn't even take a holiday. 1847 in the autumn vacation, Riemann discovered several Journals of the Paris Academy of Sciences, which contained a new paper on analytic functions of simple complex variables published by mathematician Cauchy. He saw at a glance that this was a new mathematical theory, so he stayed indoors for several weeks, devoted himself to studying Cauchy's paper and brewing his new views on this subject, laying the foundation for writing his doctoral thesis "General Theory of Functions of Simple and Complex Variables" four years later.
Riemann not only carefully studied the master's academic monographs, but also humbly asked the master for advice. Once, Dirichlet came to G? ttingen for a holiday. Riemann took this opportunity to ask him questions about mathematics and handed him his unfinished paper for advice. Dirichlet was fascinated by Riemann's modesty, sincerity and genius. He had a long talk with Riemann for two hours, made a lot of comments on Riemann's paper, and gave many directions to the topic that Riemann was studying. Riemann benefited a lot. He said that without Dirichlet's guidance, he would have to study hard in the library for a few days.
Although he lived in poverty, he studied very hard, which made Riemann achieve fruitful results when he graduated from college. 185 1 At the end, Riemann submitted his doctoral thesis to the great mathematician Gauss for review. Gauss was very excited after reading the paper and spoke highly of Riemann's paper, which is rare for Gauss. Gauss commented: "The paper submitted by Mr. Riemann provides convincing evidence that the author has conducted a comprehensive and in-depth study on this issue discussed in this paper, indicating that the author has a creative, active and true mathematical mind and brilliant creativity."
Strive for progress in poverty
At the beginning of 1852, Riemann obtained his doctorate with excellent academic performance and stayed at the University of G? ttingen. In Germany in the middle of19th century, science had almost nothing to do with the national economy. Universities are only established to train lawyers, doctors, teachers and missionaries, and provide places for aristocratic children and rich children to spend attractive and respected years. Only full professors can get government subsidies, and they can teach formal standard courses. These courses are all basic subjects, and there are many students in the class, and the professors charge a lot of tuition. This is also the reason why the course standard was low at that time, because if the course was too difficult, it would be impossible to accept many students, which would affect the professor's income. After all, the purpose of aristocratic children and rich children going to college is not to really study. Lecturers, on the other hand, have no government subsidies and no chance to teach basic formal courses. They live entirely on the tuition fees of students who come to attend classes. There are not many students attending classes at ordinary times, and their income is quite meager, so life is very difficult. Being a lecturer is the only way to become a full professor. But when the lecturer can be promoted to professor, there is no explicit provision. In order to take care of a particularly valuable scholar and there is no vacancy for a full professor, the government can appoint him as a "visiting professor" to make him qualified to teach basic formal courses and increase his income, but this appointment has additional conditions, stipulating that the government will not pay any allowance. Therefore, during his tenure as a lecturer, Riemann did not have any independent source of living expenses, and his life was still poor.
However, despite living in poverty, Riemann still devoted all his energy to mathematics. He feels that as long as he can barely make a living and let him learn mathematics, he will be satisfied. He has never been depressed by financial evidence. On the one hand, he actively prepared the inaugural speech paper of "unpaid lecturer", on the other hand, he seriously engaged in the research work of mathematical physics. His inaugural thesis is quite difficult. At first, in order to determine the topic of the paper, he submitted three topics to Gauss, who asked him to choose one. Among them, the third topic is related to the geometric basis. Riemann didn't have much desk preparation at that time, so Riemann sincerely hoped that Gauss wouldn't choose. However, Gauss studied the third topic deeply, and he thought about it for 60 years. In order to see what kind of creative work Riemann will do on this profound issue, Gauss designated the third topic as the title of Riemann's inaugural speech.
Afterwards, Riemann told his father about it, "so I fell into a desperate situation again" and "I had to create this topic".
Riemann also has infinite enthusiasm for the study of mathematical physics. At that time, he once said to people, "I am fascinated by mathematical research that combines everything with the laws of physics." "Through the general study of the relationship between electricity, light and magnetism, I found an explanation for this phenomenon. This is very important to me, because this is the first time I can apply my work to unknown phenomena. " These two studies were of high level at that time, so they were extremely difficult. Despite poverty and malnutrition, Riemann forgot his work and thought nervously for a long time, so that he was often exhausted and even fell ill. Once he has recovered a little, he will continue his research. Where there is a will, there is a way. 1854, 10 In June, Riemann delivered his inaugural speech with his paper "Assumptions on Geometric Basis", which was recognized and praised by mathematicians attending the meeting. Gauss was surprised after hearing this. He felt that the young man had handled this difficult problem well and was full of praise. Riemann's paper is regarded as one of the masterpieces in the history of mathematics in19th century.
From 65438 to 0855, the University of G? ttingen began to pay Riemann's salary, but it was quite low. A year is only equivalent to $200. In this year, Mann was 29 years old, and his family suffered great misfortune. His father and one sister died one after another, and the three sisters who once relied on their father lost their source of livelihood. So riemann sum and his brothers took on the burden of taking care of the lives of the three sisters. Riemann is always worried about the life of his family. 1857, Riemann's annual salary increased to the equivalent of $300. Because of the low income and the heavy burden of taking care of three sisters, Riemann didn't even dare to consider his marriage. However, in this year, unfortunately, it fell from the sky again, and Riemann's brother died again. This is like adding insult to injury to Riemann, and the burden of taking care of the lives of the three sisters falls on his shoulders alone. During the five years from 1855 to 1859, Riemann was always trapped by economic difficulties and poverty, and sometimes his family even fell into a situation where they needed to calculate their own rations. It is in this situation that Riemann, regardless of the poverty of material life, still devoted himself to mathematical research, struggled hard on the rugged scientific road and made amazing achievements. Many of his important achievements in mathematics were completed during this period. His research on Abel integral and Abel function initiated modern algebraic geometry. He pioneered the study of number theory with complex analytic functions and the analytic number theory in the modern sense; His research on hypergeometric series promoted the development of mathematical physics and differential equation theory. With the advent of research results, Riemann's academic reputation in the field of mathematics has rapidly increased. He was praised by many world-famous mathematicians and won the highest honor that a scientist can usually get.
The death of the master
1859 When Riemann was 33 years old, Gauss died. He was appointed as a full professor at the University of G? ttingen and became the second successor of Gauss after Dirichlet. At this time, Riemann's life began to improve, and he began to think about personal marriage. At the age of 36, he married a friend's sister. A year later, his daughter was born in Pisa.
However, the long-term poverty, overwork and vigorous research made Riemann weak and tired. 1862, Riemann suffered from pleurisy, lung disease soon, and jaundice a year later. Despite his illness, Riemann persisted in his mathematical research as long as he had a little strength. Although Riemann actively sought medical treatment during this period, he was blind due to illness and ultimately had no effect. 1866 On July 20th, Riemann's pure and noble heart stopped beating. He died prematurely, and he also left mathematics prematurely at the age of 40.
Riemann is one of the most original mathematicians in the history of mathematics. He has done a lot of basic and creative research work in many fields of mathematics: he initiated the theory of complex variable function from the geometric direction; Is the founder of analytic number theory in the modern sense; He personally established Riemannian geometry and was a pioneer of combinatorial topology. He made an important contribution to the strict treatment of calculus; It has also achieved fruitful results in the fields of mathematical physics, differential equations and so on. 1859, riemann was elected as an academician of the Berlin institute of communication; 1866, he was elected as an academician of the Paris institute of communication and a member of the royal society abroad.
Riemann's untimely death is a pity for German mathematics and even the whole world! However, in his few published papers, there are too many rich concepts that have not been fully studied by later mathematicians.
Gauss
Including people [1] and physical units [2]
[1] person:
C.F.Gauss (karl friedrich Gao? 1777.4.30 ~1855.2.23), a German mathematician, physicist and astronomer, was born in a poor family in Zwick, Germany. His father, Gerhard Di Drich, worked as a berm, bricklayer and gardener. His first wife lived with him for more than 65,438+00 years and died of illness, leaving him no children. Diderich later married Luo Jieya, and the next year their child Gauss was born, which was their only child. My father is extremely strict with Gauss, even a little too strict. He often likes to plan his life for the young Gauss according to his own experience. Gauss respected his father and inherited his honest and cautious character. De Derrick died in 1806, when Gauss had made many epoch-making achievements.
In the process of growing up, young Gauss mainly paid attention to his mother and uncle. Gauss's grandfather was a stonemason who died of tuberculosis at the age of 30, leaving two children: Gauss's mother Luo Jieya and his uncle Flier. Flier Ritchie is smart, enthusiastic, intelligent and capable, and devoted himself to the textile trade with remarkable achievements. He found his sister's son clever, so he spent part of his energy on this little genius and developed Gauss's intelligence in a lively way. A few years later, Gauss, who was an adult and achieved great success, recalled what his uncle had done for him and felt that it was crucial to his success. He remembered his prolific thoughts and said sadly, "We lost a genius because of the death of our uncle". It is precisely because Flier Ritchie has an eye for talents and often persuades her brother-in-law to let her children develop into scholars that Gauss didn't become a gardener or a mason.
In the history of mathematics, few people are as lucky as Gauss to have a mother who strongly supports his success. Luo Jieya got married at the age of 34 and was 35 when she gave birth to Gauss. He has a strong personality, wisdom and sense of humor. Since his birth, Gauss has been very curious about all phenomena and things, and he is determined to get to the bottom of it, which is beyond the scope allowed by a child. When the husband reprimands the child for this, he always supports Gauss and resolutely opposes the stubborn husband who wants his son to be as ignorant as he is.
Luo Jieya sincerely hopes that his son can do something great and cherish Gauss's talent. However, he was afraid to put his son into mathematics research that could not support his family at that time. /kloc-when she was 0/9 years old, although Gauss had made many great achievements in mathematics, she still asked her friend W. Bolyai (the father of J. Bolyai, one of the founders of non-Euclidean geometry): Will Gauss have a future? W Bolyai said that her son would become "the greatest mathematician in Europe", and her eyes were filled with tears.
At the age of seven, Gauss went to school for the first time. Nothing special happened in the first two years. 1787 years old, Gauss 10. He entered the first math class. Children have never heard of such a course as arithmetic before. The math teacher is Buttner, who also played a certain role in the growth of Gauss.
According to a story widely circulated all over the world, Gauss's most famous story is that when he was ten years old, the primary school teacher gave an arithmetic problem: "Calculate 1+2+3 …+ 100 =?" . This is difficult for beginners of arithmetic, but Gauss solved the answer in a few seconds. He used the symmetry of arithmetic progression (arithmetic progression) and then put the numbers together like a general arithmetic progression sum: 1+ 100, 2+99, 3+98, ... 49+52. However, this is probably an untrue legend. According to the research of E·T· Bell, a famous mathematical historian who has studied Gauss, Butner gave the children a more difficult addition problem: 81297+81495+81693+…+100899.
Of course, this is also a summation problem of arithmetic progression (the tolerance is 198 and the number of items is 100). As soon as Butner finished writing, Gauss finished the calculation and handed in the small tablet with the answers written on it. E. T. Bell wrote that in his later years, Gauss often liked to talk about this matter with people, saying that only his answer was correct at that time, and all the other children were wrong. Gauss didn't specify how he solved the problem so quickly. Mathematical historians tend to think that Gauss had mastered arithmetic progression's summation method at that time. For a child as young as 10, it is unusual to discover this mathematical method independently. The historical facts described by Bell according to Gauss's own account in his later years should be more credible. Moreover, it can better reflect the characteristics that Gauss paid attention to mastering more essential mathematical methods since he was a child.
Gauss's computing ability, mainly his unique mathematical methods and extraordinary creativity, made Butner sit up and take notice of him. He specially bought Gauss the best arithmetic book from Hamburg and said, "You have surpassed me, and I have nothing to teach you." Then Gauss and Bater's assistant Bater established a sincere friendship until Bater died. They studied together and helped each other, and Gauss began real mathematics research.
1788, 1 1 year-old gauss entered a liberal arts school. In his new school, all his classes are excellent, especially classical literature and mathematics. On the recommendation of Bater and others, the Duke of zwick summoned Gauss, who was 14 years old. This simple, clever but poor child won the sympathy of the Duke, who generously offered to be Gauss' patron and let him continue his studies.
Duke Brunswick played an important role in Gauss's success. Moreover, this function actually reflects a model of scientific development in modern Europe, indicating that private funding was one of the important driving factors for scientific development before the socialization of scientific research. Gauss is in the transition period of privately funded scientific research and socialization of scientific research.
1792, Gauss entered Caroline College in Brunswick for further study. 1795, the duke paid various expenses for him and sent him to the famous University of G? ttingen in Germany, which made Gauss study hard and started creative research according to his own ideals. 1799, Gauss finished his doctoral thesis and returned to his hometown of Brunswick-Zwick. Just when he fell ill because he was worried about his future and livelihood-although his doctoral thesis was successfully passed, he was awarded a doctorate and obtained a lecturer position, but he failed to attract students, so he had to go back to his hometown and the duke gave him a helping hand. The Duke paid for the printing of Gauss's long doctoral thesis, gave him an apartment, and printed Arithmetic Research for him, so that the book could be published in 180 1. Also bear all the living expenses of Gauss. All this moved Gauss very much. In his doctoral thesis and arithmetic research, he wrote a sincere dedication: "To Dagong" and "Your kindness relieved me of all troubles and enabled me to engage in this unique research".
1806, the duke was killed while resisting the French army commanded by Napoleon, which dealt a heavy blow to Gauss. He is heartbroken and has long been deeply hostile to the French. The death of Dagong brought economic difficulties to Gauss, the misfortune that Germany was enslaved by the French army, and the death of his first wife, all of which made Gauss somewhat disheartened, but he was a strong man and never revealed his predicament to others, nor did he let his friends comfort his misfortune. It was not until19th century that people knew his state of mind at that time when sorting out his unpublished mathematical manuscripts. In a discussion of elliptic functions, a subtle pencil word was suddenly inserted: "For me, it is better to die than to live like this."
The generous and kind benefactor died, and Gauss had to find a suitable job to support his family. Because of Gauss's outstanding work in astronomy and mathematics, his fame spread all over Europe from 1802. The Academy of Sciences in Petersburg has continuously hinted that since Euler's death in 1783, Euler's position in the Academy of Sciences in Petersburg has been waiting for a genius like Gauss. When the Duke was alive, he strongly discouraged Gauss from going to Russia. He is even willing to raise his salary and set up an observatory for him. Now, Gauss is facing a new choice in life.
In order not to lose Germany's greatest genius, B.A. von von humboldt, a famous German scholar, joined other scholars and politicians to win Gauss the privileged positions of professor of mathematics and astronomy at the University of G? ttingen and director of the G? ttingen Observatory. 1807, Gauss went to Kottingen to take office, and his family moved here. Since then, he has lived in G? ttingen except for attending a scientific conference in Berlin. The efforts of Humboldt and others not only made the Gauss family have a comfortable living environment, but also enabled Gauss himself to give full play to his genius, and created conditions for the establishment of Gottingen Mathematics School and Germany to become a world science center and mathematics center. At the same time, it also marks a good beginning of scientific research socialization.
Gauss's academic position has always been highly respected by people. He has the reputation of "prince of mathematics" and "king of mathematicians" and is considered as "one of the three (or four) greatest mathematicians in human history" (Archimedes, Newton, Gauss or Euler). People also praised Gauss as "the pride of mankind". Genius, precocity, high yield, persistent creativity, ..., almost all the praises in the field of human intelligence are not too much for Gauss.
Gauss's research field covers all fields of pure mathematics and applied mathematics, and has opened up many new fields of mathematics, from the most abstract algebraic number theory to intrinsic geometry, leaving his footprints. Judging from the research style, methods and even concrete achievements, he is the backbone of 18- 19 century. If we imagine mathematicians in the18th century as a series of high mountains, the last awe-inspiring peak is Gauss; If mathematicians in the19th century are imagined as rivers, then their source is Gauss.
Although mathematical research and scientific work did not become an enviable career at the end of 18, Gauss was born at the right time, because the development of European capitalism made governments around the world pay attention to scientific research when he was close to 30 years old. With Napoleon's emphasis on French scientists and scientific research, Russian czars and many European monarchs began to look at scientists and scientific research with new eyes. The socialization process of scientific research is accelerating and the status of science is improving. As the greatest scientist at that time, Gauss won many honors, and many world-famous scientists regarded Gauss as their teacher.
1802, Gauss was elected as an academician of communication and a professor of Kazan University by the Academy of Sciences in Petersburg, Russia. 1877, the Danish government appointed him as a scientific adviser, and this year, the government of Hanover, Germany also hired him as a government scientific adviser.
Gauss's life is a typical scholar's life. He has always maintained the frugality of a farmer, making it hard to imagine that he is a great professor and the greatest mathematician in the world. He was married twice, and several children annoyed him. However, these have little influence on his scientific creation. After gaining a high reputation and German mathematics began to dominate the world, a generation of Tianjiao completed the journey of life.
In the photo processing software photoshop, there is a menu called Gaussian blur, which is very useful for blurring some unnecessary places. Gauss 1777~ 1855 was born in Brunswick, north-central Germany. His grandfather is a farmer, his father is a mason, his mother is a mason's daughter, and he has a very clever brother, Uncle Gauss. He takes good care of Gauss and occasionally gives him some guidance, while his father can be said to be a "lout" who thinks that only strength can make money, and learning this kind of work is useless to the poor.
Gauss showed great talent very early, and at the age of three, he could point out the mistakes in his father's book. At the age of seven, I entered a primary school and took classes in a dilapidated classroom. Teachers are not good to students and often think that teaching in the backcountry is a talent. When Gauss was ten years old, his teacher took the famous "from one to one hundred" exam and finally discovered Gauss's talent. Knowing that his ability was not enough to teach Gauss, he bought a deep math book from Hamburg and showed it to Gauss. At the same time, Gauss is familiar with bartels, a teaching assistant who is almost ten years older than him. bartels's ability is much higher than that of the teacher. Later, he became a university professor, giving Professor Gauss more and deeper mathematics.
Teachers and teaching assistants went to visit Gauss's father and asked him to let Gauss receive higher education. But Gauss's father thought that his son should be a plasterer like him, and there was no money for Gauss to continue his studies. The final conclusion is-find a rich and powerful person to be his backer, although I don't know where to find it. After this visit, Gauss got rid of weaving every night and discussed mathematics with Bater every day, but soon there was nothing to teach Gauss in Bater.
1788, Gauss entered higher education institutions despite his father's opposition. After reading Gauss's homework, the math teacher told him not to take any more math classes, and his Latin soon surpassed the whole class.
179 1 year, Gauss finally found a patron-the Duke of Brunswick, and promised to help him as much as possible. Gauss's father had no reason to object. The following year, Gauss entered Brunswick College. This year, Gauss was fifteen years old. There, Gauss began to study advanced mathematics. Independent discovery of the general form of binomial theorem, quadratic reciprocity law in number theory, prime number theorem and arithmetic geometric average.
1795 gauss enters gottingen (g? Ttingen) university, because he is very talented in language and mathematics, so for some time he has been worried about whether to specialize in classical Chinese or mathematics in the future. At the age of 1796 and 17, Gauss got an extremely important result in the history of mathematics. It was the theory and method of drawing regular heptagon ruler that made him embark on the road of mathematics.
Mathematicians in the Greek era already knew how to make a positive polygon of 2m×3n×5p with a ruler, where m is a positive integer and n and p can only be 0 or 1. However, for two thousand years, no one knew the regular drawing of regular heptagon, nonagon and decagon. Gauss proved that:
If and only if n is one of the following two forms, you can draw a regular n polygon with a ruler:
1、n = 2k，k = 2，3，…
2, n = 2k × (product of several different Fermat prime numbers), k = 0, 1, 2, …
Fermat prime number is a prime number in the form of Fk = 22k. For example, F0 = 3, F 1 = 5, F2 = 17, F3 = 257 and F4 = 65537 are all prime numbers. Gauss has used algebra to solve geometric problems for more than 2000 years. He also regarded it as a masterpiece of his life and told him to carve the regular heptagon on his tombstone. But later, his tombstone was not engraved with a heptagon, but with a 17 star, because the sculptor in charge of carving thought that a heptagon was too similar to a circle, so people would be confused.
1799, Gauss submitted his doctoral thesis and proved an important theorem of algebra:
Any polynomial has (complex) roots. This result is called "Basic Theorem of Algebra".
In fact, many mathematicians think that the proof of this result was given before Gauss, but none of them is rigorous. Gauss pointed out the shortcomings of previous proofs one by one, and then put forward his own opinions. In his life, he gave four different proofs.
180 1 year. At the age of 24, Gauss published "Problem Arithmetic AE" written in Latin. There were eight chapters originally, but he had to print seven chapters because of lack of money. This book is all about number theory except the basic theorem of algebra in Chapter 7. It can be said that it is the first systematic work on number theory, and Gauss introduced the concept of "congruence" for the first time. "Quadratic reciprocity theorem" is also among them.
At the age of 24, Gauss gave up the study of pure mathematics and studied astronomy for several years.
At that time, the astronomical community was worried about the huge gap between Mars and Jupiter, and thought that there should be planets between Mars and Jupiter that had not been discovered. 180 1, Italy