Mathematicians have different eyes from ordinary people: mathematicians may look at problems that ordinary people find complicated and difficult very simply; Ordinary people think it is quite simple, and mathematicians may think it is very complicated. Starting with the familiar problems of middle school students, Academician Zhang Jingzhong vividly introduced how mathematicians found and drew extraordinary conclusions from these simple problems.
Through a series of "simple questions" familiar to middle school students, Eye of the Mathematician shows how mathematicians gradually deepen, analyze and dig out profound laws with wide application from these common and well-known facts. Make readers understand mathematicians' thoughts and ways of doing things and seeing problems. At the same time, it shows that mathematics is profound and thorough, which can reach the point that general discussion can't; It also shows mathematicians' pursuit of truth. Let readers understand and slowly learn the ideas and methods to solve mathematical problems in a relaxed and interesting situation.
I have read Mr. Zhang Jingzhong's articles and books for a long time, especially the words he wrote under the pseudonym "Jing". However, the first time I met Mr. Zhang was in 1989, when I was invited by the Sichuan Mathematics Society to teach a training class for mathematics Olympic teachers in Emei Mountain. In his spare time, he listened to a class of teacher Zhang. He told the primary school teacher that "chickens and rabbits live in the same cage" and was deeply impressed. He did have "Aha, a brainwave!" Sense, processing method is popular and wonderful.
Mr. Zhang's experience is not simple. He is high flyers from Peking University. When he was transferred to Xinjiang, he worked as a middle school teacher, taught a juvenile class at China University of Science and Technology, and served as a coach of the national team of the Mathematical Olympics ... Perhaps it is his profound mathematical foundation and this experience that makes him one of the famous mathematicians who know best and care most about mathematics education in primary and secondary schools in China. Mr. Zhang is an academician of China Academy of Sciences and chairman of China Popular Science Writers Association.
In addition to his busy scientific research work, he has written a large number of mathematical works that teenagers love to see. The Album of Academician's Mathematics Lecture published by China Children's Publishing House should be his masterpiece. Won the national bestseller award, the first prize of national excellent popular science works, the sixth national book award and the ninth "Five One Projects" award. In 2004, it was selected into the first batch of 100 excellent books recommended by the General Administration of Press and Publication to teenagers all over the country.
Mathematicians form a group because they have the same thinking habits. Teacher Zhang calls this "mathematician's vision", which is a good formulation, very equal and easy to accept. The difference between mathematicians and ordinary people lies in this different vision and perspective, not anything else. One of the purposes of setting up mathematics courses in primary and secondary schools is to provide students with an opportunity and environment to understand and appreciate mathematicians' vision, and teachers should be aware of this.
Through a series of "simple questions" familiar to middle school students, Eye of the Mathematician shows how mathematicians gradually deepen, analyze and dig out profound laws with wide application from these common and well-known facts. Make readers understand mathematicians' thoughts and ways of doing things and seeing problems. At the same time, it shows that mathematics is profound and thorough, which can reach the point that general discussion can't; It also shows mathematicians' pursuit of truth. Let readers understand and slowly learn the ideas and methods to solve mathematical problems in a relaxed and interesting situation.
Mr. Zhang has been at the forefront of scientific research and has made outstanding contributions to the establishment of machine-readable proof theory of geometric theorems. What is commendable is that he is good at introducing his thoughts and methods in research work in a popular and vivid way and conveying them to more people. The theoretical basis of mechanical proof of geometric theorem is "elimination point method", which is called area for short. Geometric buildings are made up of beautiful huts. Euclid chose an entrance and a path through each hut. In "New Concept Geometry", Mr. Zhang tried to take everyone to choose another entrance, take a walk and stroll in another way.
From his works, we can see that Mr. Zhang has a special liking for plane geometry, and he can also see his unique views when sorting out the geometric system. Twenty years ago, Mr. Zhang put forward the "area method" to deal with plane geometry problems. Now this method has been mastered by many middle school teachers and students, especially in solving mathematical olympiad problems. The significance of plane geometry in human rational thinking training is unique, which is a bit like physical training in sports. Table tennis players should repeatedly practice practical basic skills such as serving, catching, chopping and pumping, but they should also spend a lot of time practicing useful kung fu such as weightlifting, running and endurance, which is not so "immediate". Only with good physical fitness can they play their own level and play a good game.
We should sincerely thank Mr. Zhang for his books and his work in popularizing mathematics. I really hope that more "Zhang Jingzhong" will care about, support and practice this matter. There are several martin gardner-style figures in China!
Others:
Topic: Discrete Mathematics (1)
Textbook of Tsinghua University Computer Department
Discrete mathematics is the core course of basic theory of computer science. It includes mathematical logic, set theory, algebraic structure, graph theory, formal language, automata and computational set.
The first chapter is the basic concept of propositional logic.
Section 1 Proposition
First, what is a proposition?
A proposition is a true or false statement.
1) Proposition is a declarative sentence.
2) The contents expressed in this statement are either true or false.
We turn this propositional logic into binary logic, and turn the logic taking this proposition as the research object into classical logic.
Second, propositional variables
We agree to use uppercase letters to represent propositions and lowercase letters to represent propositional variables. Proposition refers to a specific statement with a definite truth value; However, the true value of propositional variables is uncertain. Only when a specific proposition is substituted into the propositional variable and the propositional variable is transformed into a proposition can its true value be determined.
Three, simple proposition and compound proposition
A proposition that cannot be decomposed into a simpler combination of propositions is called a simple proposition. Also called atomic proposition, it does not contain any conjunctions such as AND, OR and NOT.
A proposition in which one or several simple propositions are connected by conjunctions (such as AND, OR, NO) is called a compound proposition, also called a molecular proposition.
Section 2 Propositional Connectors and Truth Tables
Connectives fall into two categories:
1) truth-value connectives, and the truth value of the compound proposition composed of these connectives is completely determined by the truth value of the simple proposition that constitutes it.
2) Non-truth connectives, the truth value of a compound proposition is not completely determined by the truth value of the simple proposition that constitutes it.
First, the negative word┑
The negative ┑ is a unary conjunction. A proposition P with negative words constitutes a new proposition. Written as ┑P, this new proposition is the negation of proposition P, pronounced as non-P.
The truth values of proposition P and proposition non-P are different from each other.
Second, the conjunction∧
The conjunction "∧" is a binary propositional conjunction. Conjunction connects two propositions P and Q to form a new proposition P∧Q, which can be read as a conjunction of P and Q, and P and Q can be simple propositions or compound propositions.
P and q are true only if they are both true, otherwise they are false.
Namely:
P=T
Q=T
P∧Q=T
Third, the turning point∨
The disjunctive word "∨" is a binary proposition conjunction word, which connects two propositions P and Q to form a new proposition P∨Q, which can be read as disjunctive of P and Q, or as P or Q. 。
P∨Q is false only if both P and Q are false (F), otherwise P∨Q is true.
Namely:
P=F
Q=F
p∞Q = F
Four. Implied words →
The implication word "→" is also a binary propositional conjunction, which connects two propositions P and Q to form a new proposition P→Q, which is read as if P is Q, or as if P contains Q, and as if P is Q, where P is the antecedent (the previous paragraph, the condition) and Q is the latter (the latter item, the conclusion).
It is stipulated that only when P is true and Q is false, P→Q=F, otherwise P→ Q = T..
Namely:
P=T
Q=F
P→Q=F
Under P→Q=T, if P=T, there must be Q=T, which shows that P→Q embodies the sufficient condition that p is q.
Under P→Q, if P=F, there can be Q=T, which shows that P→Q embodies the necessary condition that p is not necessarily Q.
Truth table of P→Q
P Q P→Q
F F T
F T T
T F F
T T T
The truth table of ┑P∨Q
┑ P ? Q Oil Company
F F T
F T T
T F F
T T T
At all values of p and q, P→Q and ┑P∨Q have the same truth value.
Namely: P→Q=┑P∨Q
Equivalent propositions with the same truth value are connected by an equal sign. This shows that → can be expressed by ┑ and ∨. Logically, "If P is Q" and "Neither P nor Q" are two equal propositions.
Verb (abbreviation of verb) double conditional word =
The double conditional word "=" (represented by a number and a pair of arrows in some books) is also a binary proposition conjunction, which connects two propositions P and Q to form a new proposition P=Q, which is pronounced as P if and only if Q or P is equivalent to Q. 。
Only when the truth values of two propositions P and Q are the same, the truth value of P=Q is T.
Truth table of P=Q
P Q P=Q
F F T
F T F
T F F
T T T
Section III Combination Formula (Formula for short)
Definition of combination formula:
1. Simple proposition is a compound formula.
2. If A is a formula, then ┑A is also a formula.
3. If A and B are combined formulas, then (A∧B), (A∨B), (A→B) and (A=B) are also combined formulas.
4. The symbol string composed of 1, 2,3 is a compound formula if and only if it is used a limited number of times.
Agreed conjunctions are arranged in the order of ┑, ∨, ∧, →, =.
Section 4 tautology
I. Definition
There is a tautology in the propositional formula. If a formula is true for any of its explanations, it is called tautology (eternal truth). For example, P∨┑P is a tautology.
Obviously, the tautology connected by ∨, ∧, →, = is still tautology.
If there is an explanation for I0, and the true value of the formula is true under I0, the formula is said to be satisfiable.
If the truth value of a formula is false for any of its explanations I, then it is called forever false (contradictory) or unsatisfiable. For example, P∧┑P is an oxymoron.
The relationship between these three formulas:
1. Formula A is always true if and only if ┑A is always false.
2. Formula A can be satisfied if and only if ┑A is not always true.
3. Unsatisfactory formulas must always be false.
4. Formulas that cannot be false forever must be satisfied.
Second, the substitution rule.
A is a formula. For a, use formula b of the replacement rule. If A is tautology, then B is tautology.
In order to ensure that tautology remains after being substituted into rules, it is required that:
1. Formula can only replace atomic propositions, but not compound propositions.
2. To replace a propositional variable in a formula, all the same propositional variables in the formula must replace the same formula.
Section 5 Formalization of Simple Natural Sentences
First, the formalization of simple natural sentences
Second, the formalization of more complex natural sentences.
Section 6 Polish expression
First of all, the process of computer recognizing parentheses
In the definition of combinatorial formulas, infixes of conjunctions are used, and brackets are introduced to distinguish the operation order. These are common methods.
The method of computer identifying and processing the formula expressed in this way requires repeated scanning from left to right and from right to left. Such as pairing formula
(P∨(Q∧R))∨(S∧T)
The calculation process of the truth value starts from left to right until the first right bracket is found, and then returns to the nearest left bracket. You can use part of formula (Q∧R) to calculate the truth value, and then scan to the right until the second right bracket is found, and then return to the second left bracket, so you can get part of formula (P∨(Q∧R).
Second, Polish style.
Generally speaking, there are three ways to use conjunctions to form formulas: infix type such as P∨Q, prefix type such as PQ, and suffix type such as PQ∨.
The use of prefix in logic was put forward by J. Lukasiewicz, a Polish mathematical logician, and called it Polish expression.
If the expression of formula (P ∪( Q∧R))∪( S∧T) is changed to Polish, the brackets can gradually separate from the outer layer (or from the outer layer to the inner layer).
Polish expression of formula (P∨(Q∧R))∨(S∧T);
∨P∧∨QRS
Polish expression formula, the process of computer recognition processing can be completed at one time when scanning from right to right, avoiding repeated scanning. The same backward expression (anti-Polish expression) also has the same advantages. Scanning from left to right (which looks more reasonable) is easy to identify and process a formula, which is often used by computer program systems, but people are not used to reading this expression formula.
Mathematics series "
There are many classic popular mathematics books in China, some of which have been handed down from generation to generation. Unfortunately, most of them have a small print run, basically no more than 5,000, and some classics are no longer published, making it difficult for people who like mathematics to find books.
A very gratifying thing in recent years is that the Mathematics Series published in 1960s and written by famous mathematicians was republished by Science Press in 2002.
In this series of pamphlets 18, Hua wrote five books-from Yang Hui Triangle, from Zu Chongzhi's pi, from Sun Tzu's "divine calculation", from mathematical induction, and from mathematical problems related to honeycomb structure, all of which were brilliant and carefully chosen. Hua Lao's popular science articles have a major feature, that is, creativity. In this popular science essay, he can still have his own unique thinking on some issues. For example, Li's proof of identities in mathematical induction. There is a story circulating here: in the early 1950s, Paul Turán, a famous Hungarian mathematician, visited China and gave a report at the Institute of Mathematics in China. In the report, he gave the proof of Li's identity, which was a mathematical discovery made by mathematicians in the late Qing Dynasty. This theorem was discovered by China people, but it was not proved by China people. As a mathematician in China, Hua has a strong sense of national pride. Back to his residence, he thought hard and finally gave his identity certificate again before dawn. Early in the morning, when he said goodbye to Paul Duran, he gave Duran a note. When Duran looked at it, he found that it was Hua's concise proof of Li Heng's identity, which was very elementary and beautiful compared with the proof that he needed some advanced mathematics! I don't know how Turan reacted at that time, but I think I have to admire the wisdom of China people at least.
Now Academician Zhang Jingzhong has inherited the style of this popular science article. His eyes of mathematicians (2007 supplementary edition) made a very unique thinking on the basis of calculus. This book was highly praised by some mathematicians and even appreciated by Chen Shengshen. In a letter to Zhang Jingzhong, Chen Shengshen suggested that the book be translated into a foreign language and published. Other popular mathematics books in Zhang Jingzhong are equally wonderful, such as Help You Learn Mathematics, Random Talk on Mathematics, Miscellaneous Talk on Mathematics, Starting from Root Number 2, New Concept Geometry, From Mathematics Education to Educational Mathematics, Mathematics and Philosophy, etc. These books have been compiled into Academicians' Lectures on Mathematics and published by China Children's Publishing House. Zhang Jingzhong also edited a set of interesting mathematics, some of which are very suitable for primary and junior high school students.
Hua's pamphlet has a great influence. When Qiu Chengtong was studying mathematics in middle school, he benefited from these popular science books of Hua Lao. The reporter of Science Times described in Qiu Chengtong: Young Students Should Cultivate the Attitude of Learning for Learning: Because of his poor family, Qiu Chengtong could not afford books when he was in middle school, so he went to libraries and bookstores to read books. Mathematician Hua's books benefited him a lot: "At that time, we had few books, mainly reading books published in mainland China, because mainland books were very cheap, and I read at least 15 of Mr. Hua's books, such as". I have also read some pamphlets written by Chen Mingzhe. Therefore, I finished all the exercises one semester ahead of the course, and listening to math class became a pleasure. Hua's pamphlets and some of his articles were compiled into Selected Works of Hua Popular Science, which was published by Shanghai Education Publishing House in 1980s. Recently, it has been divided into two volumes: cleverness lies in diligence and genius lies in accumulation: Master Hua talks about how to learn mathematics well, which comes from Sun Tzu's Art of War, Divine Calculation: A Gift from Master Hua to Middle School Students, reprinted by China Children's Publishing House. But some chapters do not include, such as very delicate finite and infinite, discrete and continuous.
Regarding how to learn mathematics, I personally think that Hua's "cleverness lies in diligence and genius lies in accumulation" is the best choice. Hua is self-taught, and he has a unique method on how to study and research. Although his articles are branded with some times, apart from those political things, I personally think they can be called the Bible for learning mathematics. China People's Liberation Army Publishing House republished a book with the same content, named "Chinese Chess Manual".
There are Wu Wenjun's Some Applications of Mechanics in Geometry, Duan Xuefu's Symmetry, Shi Jihuai's Average, Min Sihe's Lattice and Area, Jiang Boju's A Post Road, Gong Sheng's Starting from Liu Hui's Secant Circle, Fan Huiguo's Several Types of Extreme Value Problems, Cai Zongxi's Equidistance Problems and Jiang Zehan's.
I noticed that these masterpieces actually need the support of Tianyuan Mathematics Foundation to be reprinted, which is very embarrassing.
I think it's two books that the reporter misremembered, Analysis of Number Theory and Introduction to Number Theory written by Hua. It should be an introduction to number theory and advanced mathematics. Before Qiu Chengtong entered the university, his mathematics level was quite high. Masters always learn directly from masters!