Geometric element
number; amount; how many; how much
what
origin
basis
The Elements of Geometry is the immortal work of Euclid, an ancient Greek mathematician. It is the crystallization of the achievements, methods, thoughts and spirit of the whole Greek mathematics at that time. Its content and form have great influence on geometry itself and the development of mathematical logic. Since its publication, it has been popular for more than 2000 years. It has been translated and revised many times. Since the first printed version of 1482 came out, there have been more than 1000 different versions so far. Apart from the Bible, no other works have been widely studied, used and spread, which can be compared with the Elements of Geometry. However, The Elements of Geometry has transcended the influence of nationality, race, religious belief and cultural consciousness, but it is incomparable to the Bible.
After the 7th century BC, Greek geometry developed rapidly and accumulated rich data. Greek scholars began to organize the mathematical knowledge at that time in a planned way and tried to form a strict knowledge system. Hippocrates first tried this method in the 5th century BC, and it was revised and supplemented by many mathematicians. By the 4th century BC, Greek scholars had laid a solid foundation for establishing a theoretical building of mathematics.
On the basis of predecessors' work, Euclid collected and sorted out the rich mathematical achievements of Greece, restated them in the form of propositions, and strictly proved some conclusions. His greatest contribution is to select a series of original definitions and axioms of great significance, arrange them in strict logical order, and then deduce and prove them on this basis, thus forming geometrical features with axiomatic structure and strict logical system.
The Greek manuscript of The Elements of Geometry has been lost, and all its modern editions are based on the revised edition written by the Greek critic Theon (about 700 years later than Euclid). There are 465 propositions in volume 13 of the revised edition of Elements of Geometry, which are about the systematic knowledge of plane geometry, solid geometry and arithmetic theory.
The first volume first gives some necessary basic definitions, explanations, postulates and axioms, and also includes some well-known theorems about congruence, parallel lines and straight lines. The last two propositions in this volume are Pythagoras theorem and its inverse theorem. Here we think of a short story about the English philosopher T. Hobbes: One day, Hobbes accidentally read Euclid's Elements of Geometry and was surprised to see Pythagoras theorem. He said: This is impossible. "He carefully read the proof of each proposition in the first chapter from back to front until he was completely convinced by axioms and postulates. The second volume is not long, mainly discussing the Pythagorean school's geometric algebra.
The third volume includes some famous theorems of circle, chord, secant, tangent, central angle and circumferential angle. Most of these theorems can be found in the current middle school mathematics textbooks. The fourth volume discusses the ruler drawing of some inscribed and circumscribed regular polygons of a given circle.
The fifth volume gives a wonderful explanation of eudoxus's proportional theory, which is considered as one of the most important mathematical masterpieces. It is said that Porzano (Porzano, 178 1- 1848), an unknown mathematician and priest in Czechoslovakia, happened to be ill when he was on vacation in Prague. In order to distract him, he picked up Geometry and read the first book. This wonderful method excited him and completely relieved his illness. Since then, whenever his friend is ill, he always takes it as a panacea and asks patients to recommend it.
The seventh, eighth and ninth volumes discuss elementary number theory, give Euclid algorithm for finding the greatest common factor of two or more integers, discuss proportion and geometric series, and give many important theorems about number theory.
The tenth volume discusses unreasonable quantities, that is, incommensurable line segments, which are difficult to read. The last three volumes, namely the eleventh, twelfth and thirteenth volumes, discuss solid geometry. At present, most of the contents in middle school geometry textbooks can be found in Geometry Elements.
According to the axiomatic structure and using Aristotle's logical method, Geometry Elements established the first complete knowledge system of geometric deduction. The so-called axiomatic structure is to select a small number of unproven original concepts and propositions as definitions, postulates and axioms, making them the starting point and logical basis of the whole system, and then proving other propositions through logical reasoning. The Elements of Geometry has become an excellent example of using axiomatic methods for more than 2000 years.
It is true that, as some modern mathematicians have pointed out, the Elements of Geometry has some structural defects, but this does not detract from the lofty value of this work. Its far-reaching influence makes "Euclid" and "geometry" almost synonymous. It embodies the mathematical thought and spirit laid by Greek mathematics and is a treasure in human cultural heritage.
Goldbach's Conjecture
brothers
morality
Eager hope/persistence
hertz
guess
think
1742, the German Goldbach wrote a letter to Euler, a great mathematician living in Petersburg, Russia. In the letter, he raised two questions: First, can every even number greater than 4 be expressed as the sum of two odd prime numbers? Such as 6 = 3+3, 14 = 3+ 1 1 and so on. Second, can every odd number greater than 7 represent the sum of three odd prime numbers? Such as 9=3+3+3, 15=3+5+7, etc. This is the famous Goldbach conjecture. This is a famous problem in number theory, which is often called the jewel in the crown of mathematics.
In fact, the correct solution of the first question can lead to the correct solution of the second question, because every odd number greater than 7 can obviously be expressed as the sum of even numbers greater than 4 and 3. 1937, the Soviet mathematician vinogradov proved that every odd number big enough can be expressed as the sum of three odd prime numbers with his original "triangular sum" method. Basically solved the second problem, the first problem has not been solved. Because the problem is too difficult, mathematicians began to study the weak proposition: every even number large enough can be expressed as the sum of two natural numbers with prime factors m and n respectively, abbreviated as "m+n" The Norwegian mathematician Braun proved "9+9" in 1938+0920; In the next 20 years, mathematicians successively proved "7+7", "6+6", "5+5", "4+4" and "1+C", where c is a constant. 1956 China mathematician Wang Yuan proved "3+4" and later. Mathematicians at home and abroad have advanced the proposition to "1+3". 1966, China mathematician Chen Jingrun proved "1+2", which is called "Chen Theorem" and is still the best result. Chen Jingrun's outstanding achievements made him widely praised, not only because "Chen Theorem" made China take the lead in proving Goldbach's conjecture. More importantly, the spirit of a large number of China mathematicians, represented by Chen Jingrun, who overcame many difficulties and bravely climbed the peak will always inspire and inspire young people with lofty ideals to strive to make China a world mathematical power in the 2 1 century!
The Influence of Computer on Mathematics
electric current
brain
correct
count
study
about
shadow
loudly
For the convenience of narration, I call the mathematics before the appearance of computers classical mathematics, and the mathematics after the appearance of computers modern mathematics.
1. Research contents and methods of classical mathematics
(1) From book thesis to book thesis: You can study mathematics with a piece of paper and a pen.
(2) only do qualitative research between quantity and quantity;
(3) A few people (mathematicians) are engaged in ivory tower research;
(4) The degree of solving mathematical problems has become an important method to measure the level of mathematical research;
(5) The number and level of papers published in mathematical journals become the only measure;
(6) Mathematics absorbs "nutrition" through other disciplines, and mathematics acts on production through other disciplines, which has an indirect effect on production;
(7) Mathematics is the foundation of other disciplines.
2. Characteristics and contents of modern mathematics
(1) is directly related to production through computer;
Absorb "nutrition" directly from production.
Direct impact on production
Mathematics plays a more important role in production than any other subject.
(3) Mathematics and computer are inseparable;
Mathematics can't be separated from computers. Without computers, there would be no modern mathematics.
Computers can't be separated from mathematics. Without mathematics, there is no computer.
Mathematics will develop with the rapid development of computers.
The development of mathematics reacts on computers, and the development of computers cannot be separated from the development of mathematics.
(3) Software is the only bridge between mathematics and computer;
Without software, there would be no modern mathematics.
Without software, the computer is just a waste.
Computer = software+hardware
(4) Modern mathematics includes the following contents:
Establishment of mathematical model
Analyze the model mathematically and demonstrate the correctness of the model from a mathematical point of view.
Selection of algorithm
The algorithm is analyzed mathematically, and the effectiveness of the model is demonstrated from a mathematical point of view.
Compilation and debugging of software
Comparison between software operation effect and mathematical analysis (theoretical result)
(5) Mathematics is not only the foundation of other disciplines.
Mathematics (combined with computer) has become the third means for human beings to understand and transform the world, breaking through the other two means.
-Theoretical and practical limitations
The combination of mathematics and computer is productivity.
(6) Mathematics is no longer a subject studied by a few people;
Everyone needs to use a computer, which is inseparable from mathematics; Mathematics has become a knowledge and tool that everyone must master.
Everyone is using mathematics, and everyone can engage in mathematical research.
Mathematics has gone far beyond the category of classical reasoning mathematics (from the Mathematics Education Forum)
Modern mathematician
at present
produce
count
study
home
1. The invention and development of computers have greatly shortened the distance between science and production, especially between mathematics and production.
(1) Mathematics has completely gone out of the "ivory tower" and become a part of products or production tools, and may even be the most important part;
(2) Take mathematics as the core.
numerical simulation
numerical simulation
Numerical test
It has become an important part of modern scientific experiment and production process.
(3) Optimal design is the highest realm of product design.
Mathematics is the soul of optimal design.
(4) The digital revolution (information revolution) is a new production revolution after industrialization, and mathematics will become the core content of this revolution.
Modern mathematicians are different from classical mathematicians. They can't just understand reasoning mathematics. They should have the following knowledge:
(1) They should be proficient in not only one branch of mathematics, but also many branches of mathematics.
(2) In addition to mathematics, we should also understand other professional disciplines and be able to communicate with engineers and experts in other disciplines.
(3) Know how to establish a correct mathematical model.
(4) Know how to use computers to solve problems.
(5) Know how to convert the algorithm into software.
(6) Know how to do mathematical reasoning and analysis on models and algorithms.
Only the last item belongs to classical mathematics, and the other five items are not within the scope of classical mathematics, but modern mathematicians must have knowledge, so the knowledge of modern mathematicians is much wider than that of classical mathematicians.
3. The mission of modern mathematicians
(1) The research results of classical mathematicians are mainly expressed in mathematical papers, so the achievements of mathematicians in the past were always measured by the number and level of published mathematical papers.
(2) But for modern mathematicians, mathematical papers are only a part of their research achievements, and they are often not their main achievements.
(3) For most modern mathematicians, their main focus should be on how to use mathematics and computers to solve various problems in science and production.
(4) The development of modern science and technology is inseparable from the development of computers and modern mathematics. Modern mathematicians who have mastered computers and mathematics are the most important and basic scientific modernization team.
(5) China cannot achieve the four modernizations and catch up with the advanced world level without this modern team of scientists. To support basic disciplines, we must first support the growth, development and expansion of this team.