The Mathematician's Eye doesn't talk about the skills of solving a certain kind of mathematical problems. It tells readers the ideas and methods of thinking about mathematical problems, focusing on helping readers improve their ability to solve mathematical problems in an all-round way. The mathematician's eyes is praised by experts at home and abroad as a masterpiece of popular science with advanced world level.
Mathematicians' eyes are different from those of ordinary people: mathematicians may find problems that are very complicated and difficult for ordinary people very simple; What ordinary people think is quite simple, mathematicians may think it is very complicated. Academician Zhang Jingzhong started with the familiar problems of middle school students, and vividly introduced how mathematicians found and drew extraordinary conclusions from these simple problems.
The Mathematician's Eye shows how mathematicians proceed from these common and well-known facts, analyze and dig out the profound laws with wide application step by step through a series of "simple questions" familiar to middle school students. Make readers understand mathematicians' ideas and methods 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 the mathematicians' pursuit of truth. So that readers can 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 his writing under the pen name "Inoue". However, the first time I met Mr. Zhang was in 1989, when I was invited by the Sichuan Mathematical Society to teach at Emei Mountain for the Mathematical Olympiad teacher training class. In his spare time, he listened to a class of Mr. Zhang. He told the primary school teachers 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 of Peking University, worked as a middle school teacher when he was transferred to Xinjiang, taught a juvenile class at the University of Science and Technology of China, and served as the coach of the national team of the Mathematical Olympics ... Perhaps it is his profound mathematical skills and this experience that make him one of the famous domestic mathematicians who know best and care most about mathematics education in primary and secondary schools. Mr. Zhang is now an academician of China Academy of Sciences and chairman of China Popular Science Writers Association.
After his busy scientific research work, he has written a lot of popular mathematics works for teenagers. The "Academician Mathematics Lecture Album" published by China Children's Publishing House should be his masterpiece. Won the national best-seller award, the first prize of national excellent popular science works, the sixth national book award and the ninth "Five One Projects" award. In 24, it was selected into the first batch of 1 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. Mr. Zhang calls this "mathematician's vision", which is a good, equal and easy to accept. The difference between mathematicians and ordinary people lies in this different vision and angle of view, not anything else. One of the purposes of offering mathematics courses in primary and secondary schools is to provide students with an opportunity and environment to understand and appreciate the mathematician's vision, and teachers should be aware of this.
The Mathematician's Eye shows how mathematicians proceed from these common and well-known facts, analyze and dig out the profound laws with wide application step by step through a series of "simple questions" familiar to middle school students. Make readers understand mathematicians' ideas and methods 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 the mathematicians' pursuit of truth. So that readers can understand and slowly learn the ideas and methods to solve mathematical problems in a relaxed and interesting situation.
Mr. Zhang has been standing at the forefront of scientific research, doing excellent work for establishing the theory of machine readability proof of geometric theorems. What is valuable 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 simply called area. Geometry building is made up of beautiful huts. Euclid chose an entrance and a path to go 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 we can see his unique views in 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, and its advantages in solving mathematical Olympic problems are particularly obvious. The significance of plane geometry in human rational thinking training is unique, which is a bit like physical training in sports. Table tennis players have to practice the practical basic skills such as serving, catching, chopping and pumping repeatedly, but they also have to spend a lot of time practicing weightlifting, running, endurance and other less "immediate" useful kung fu. Only with good physical fitness can they play their level and play a good game.
We should sincerely thank Mr. Zhang for his books and his work for the popularization of mathematics. I really hope that more "Zhang Jingzhong" will care, support and practice this matter, and several martin gardner-style figures will appear in China!
Others:
Title: discrete mathematics (I)
The textbook of Tsinghua University Computer Department
Discrete Mathematics is the core course of the basic theory of computer science. It includes mathematical logic, set theory, algebraic structure, graph theory, formal language, automata and computational set.
Chapter I Basic Concepts of Propositional Logic
Section I Proposition
I. What is a proposition
A proposition is a statement that is either true or false.
1) A proposition is a declarative sentence.
2) The content expressed in this statement is either true or false.
We turn this propositional logic into binary logic, and the logic that takes this proposition as the research object into classical logic.
2. Propositional variables
We agree to use uppercase letters to represent propositions and lowercase letters to represent propositional variables. Proposition refers to concrete statements with definite truth values; However, the truth value of propositional variables is uncertain. Only when a specific proposition is substituted into propositional variables, the propositional variables are transformed into propositions, and its truth value can be determined.
III. Simple propositions and compound propositions
A proposition that cannot be decomposed into a combination of simpler propositions is called a simple proposition. It is also called the atomic proposition, which does not contain any conjunctions such as AND, OR and NOT.
A proposition that one or several simple propositions are connected by conjunctions (such as AND, OR and NOT) is called a compound proposition, which is also called a molecular proposition.
Propositional conjunctions and truth tables in the second section
Conjunctions are divided into two categories:
1) Truth conjunctions, and the truth of the compound proposition formed by these conjunctions is completely determined by the truth of the simple proposition that constitutes it.
2) Non-truth connectives, in which the truth of a compound proposition is not completely determined by the truth of a simple proposition.
1. The negative word ┑
The negative word ┑ "is a unitary conjunction. A proposition P with negative words constitutes a new proposition. Written as ┑P, this new proposition is the negation of proposition p, and the truth and falsehood of proposition p and proposition p are different from each other.
Second, the conjunction ∧
The conjunction ∧ "is a binary propositional conjunction. Conjunctive words connect two propositions p and q to form a new proposition P∧Q, which can be read as the conjunction of p and q or as p and q. Where 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
3. disjunctive word ∨
disjunctive word "∨" is a binary propositional conjunction, which connects two propositions P and Q to form a new proposition P.
that is:
P=F
Q=F
P∨Q=F
Fourth, the implication word →
The implication word → "is also a binary proposition conjunction, which connects two propositions P and Q to form a new proposition P→Q, pronounced as if P. Where P refers to the antecedent (the preceding paragraph, the condition) and Q refers to 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
that is,
P=T
Q=F
P→Q=F
P→Q=T, if P=T, there must be Q=T.
under P→Q, if P=F, there can be Q=T, which shows that p → q embodies the necessary condition that p does not have to be q.
the truth table of P→Q
p q p→ q
f f t
f f f
t t t
p ┑P∨Q
p q ┑ p ∨ q < This shows that → can be represented by ┑ and ∨, and logically, "If P is Q" and "Not P or Q" are two equal propositions.
V. Double conditional word =
The double conditional word "=" (in some books, it is represented by the number one with double arrows) is also a binary propositional 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 the two propositions P and Q are the same, The truth table with the truth square of P=Q as T
P=Q
p q p = q
f f t
f f f
t f
t t t
the third section formula (formula for short)
definition of the combined formula:
1. Then (A∧B), (A∨B), (A→B) and (A=B) are also compound formulas
4. A compound formula is only if and only if the symbol string consisting of 1, 2 and 3 is used a limited number of times.
The agreed conjunctions are arranged in the order of ┑, ∨, ∧, →, =.
tautology in the fourth quarter
I. Definition
There is a kind of tautology in the propositional formula. If the truth of 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.
a formula is said to be satisfiable if there is an explanation I and the truth value of the formula is true under I.
if the truth value of a formula is false for any of its explanations I, it is said to be forever false (contradictory) or unsatisfiable. For example, P∧┑P is 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. Formula that is not satisfied must always be false
.
In order to ensure that tautology can still be preserved after being substituted into the rules, the following requirements are required:
1. Only atomic propositions can be substituted in the formula, not compound propositions.
2. To substitute a propositional variable in a formula, all the same propositional variables in the formula must be substituted for the same formula.
section 5 formalization of simple natural sentences
1. formalization of simple natural sentences
2. formalization of more complex natural sentences
section 6 polish expressions
1. the process of computer recognizing brackets
in the definition of a compound formula, the infix representation of conjunctions is used, and brackets are introduced to distinguish the operation order. These are common methods.
The computer needs to scan from left to right and from right to left repeatedly to identify and process the formula expressed in this way. If the calculation process of the truth value of formula
(P∨(Q∧R))∨(S∧T)
starts to scan from left to right until the first right bracket is found, then it returns to the nearest left bracket, and only part of formula (Q∧R) can be calculated.
2. Polish style
Generally speaking, there are three ways to form formulas by using conjunctions, such as infix type, P∨Q, prefix type, PQ∨
The prefix type used in logic was put forward by J. Lukasiewicz, a Polish mathematical logician, and is called Polish expression.
if the expression of formula (P∨(Q∧R))∨(S∧T) is changed into Polish style, the inner bracket can be gradually detached from the outer layer (or from the outer layer to the inner layer)
Formula (p ∨ (q ∨) The same backward expression (anti-Polish expression) also has the same advantages, and scanning from left to right (which seems more reasonable) makes it easy to identify and process a formula, which is often adopted by computer program systems, but people are not used to reading this expression formula.
Mathematics Series
There are many classic books on popular mathematics in China, some of which are handed down from generation to generation. Unfortunately, most of them have a small print run, basically no more than 5, copies, and some classics are no longer published, making it hard for people who like mathematics to find a book.
A very gratifying thing in recent years is that the Mathematics Series, which was published in the 196s and written by famous mathematicians and mathematicians, was re-published by Science Press in 22.
among the 18 booklets in this series, Hua Luogeng has written five booklets-from Yang Hui Triangle, from Zu Chongzhi's Pi, from Sun Tzu's "magical calculation", mathematical induction and talking about mathematical problems related to honeycomb structure, all of which are splendid and full of words! Hua Lao's popular science articles have a major feature, that is, creativity. In this popular science essay, he can still have his own original thinking on some issues. For example, the proof of Li Shanlan's identity in Mathematical Induction. There is a story circulating here: in the early 195s, Paul Turán, a famous Hungarian mathematician (who discovered the famous Turan theorem in graph theory), visited China and gave a report in the Institute of Mathematics where Hua Luogeng was located. In the report, he gave a proof of Li Shanlan's identity, a mathematical discovery from mathematicians in the late Qing Dynasty. This theorem was discovered by China people, but it was not proved by China people. Hua Luogeng as a mathematician in China