Several important theorems: Menelius Theorem, Seva Theorem, Ptolemy Theorem and siemsen Theorem.
Several Special Points in Triangle: Imitation Center, fermat point and Euler Line.
Geometric inequality.
Geometric extremum problem.
Transformation in geometry: symmetry, translation and rotation.
Power and root axis of a circle.
Area method, complex number method, vector method, analytic geometry method.
2. Algebra
Periodic function, a function with absolute value.
Trigonometric formula, trigonometric identity, trigonometric equation, trigonometric inequality, inverse trigonometric function.
Recursion, recursive sequences and their properties, general formulas of first-order and second-order linear recursive sequences with constant coefficients.
The second mathematical induction.
Mean inequality, Cauchy inequality, rank inequality, Chebyshev inequality, univariate convex function.
Complex number and its exponential form, triangular form, Euler formula, Dimov theorem, unit root.
Polynomial division theorem, factorization theorem, polynomial equality, rational root of integer coefficient polynomial *, polynomial interpolation formula *.
The number of roots of polynomials of degree n, the relationship between roots and coefficients, and the virtual root pairing theorem of polynomials with real coefficients.
Function iteration, simple function equation *
3. Elementary number theory
Congruence, Euclid division, Pei tree theorem, complete residue class, quadratic residue, indefinite equations and equations, Gaussian function [x], Fermat's last theorem, lattice point and its properties, infinite descent method, euler theorem *, Sun Tzu's theorem *.
4. Combination problem
Cyclic permutation, permutation and combination of repeated elements, combinatorial identity.
Combinatorial counting, combinatorial geometry.
Dove cage principle
Exclusion principle.
Extreme principle.
Graph theory problems.
Division of sets.
Cover.
Planar convex set, convex hull and their applications.
Paradox words
Hippasus Paradox and the First Mathematical Crisis
The proposition of hippasus's paradox is closely related to the discovery of Pythagorean theorem. So, let's start with Pythagorean theorem. Pythagorean theorem is one of the most famous theorems in Euclidean geometry. Astronomer Kepler once called it one of the two bright pearls in Euclidean geometry. It is widely used in mathematics and human practice, and it is also one of the earliest plane geometry theorems recognized by human beings. In China, the earliest astronomical mathematics book "Zhou Pi's Mourning Classic" has a preliminary understanding of this theorem. However, the proof of Pythagorean theorem in China was later. Until the Three Kingdoms period, Zhao Shuang used area cutting to provide the first proof.
Abroad, Pythagoras of ancient Greece first proved this theorem. Therefore, it is generally called "Pythagoras Theorem" abroad. It is also said that Pythagoras was ecstatic after completing this theorem and killed 100 cows to celebrate. Therefore, this theorem has also won a mysterious title: "Hundred Cows Theorem".
Pythagoras
Pythagoras was a famous mathematician and philosopher in ancient Greece in the fifth century BC. He once founded a school of mysticism: Pythagoras School, which integrates politics, scholarship and religion. Pythagoras' famous proposition "Everything is a number" is the philosophical cornerstone of this school. "All numbers can be expressed as integers or the ratio of integers" is the mathematical belief of this school. Dramatically, however, the Pythagorean theorem established by Pythagoras has become the "grave digger" of Pythagoras' mathematical belief. After the Pythagorean theorem was put forward, hippasus, a member of his school, considered a question: What is the diagonal length of a square with a side length of 1? He found that this length can not be expressed by integer or fraction, but only by a new number. Hippasus's discovery led to the birth of the first irrational number √2 in the history of mathematics. The appearance of small √2 set off a huge storm in the mathematics field at that time. It directly shook the Pythagorean school's mathematical belief and made the Pythagorean school panic. In fact, this great discovery is not only a fatal blow to Pythagoras school. This was a great shock to the thoughts of all the ancient Greeks at that time. The paradox of this conclusion lies in its conflict with common sense: any quantity can be expressed as a rational number within any precision range. This is a widely accepted belief not only in Greece at that time, but also in today's highly developed measurement technology. However, the conclusion that is convinced by our experience and completely in line with common sense is overturned by the existence of a small √2! How contrary to common sense and ridiculous this should be! It just subverts the previous understanding. To make matters worse, people are powerless in the face of this absurdity. This directly led to the crisis of people's understanding at that time, which led to a big storm in the history of western mathematics, known as the "first mathematical crisis."
Eudoxus
Two hundred years later, around 370 BC, the brilliant eudoxus established a complete set of proportional theory. His own works have been lost, and his achievements are kept in the fifth chapter of Euclid's Elements of Geometry. Eudoxus's ingenious method can avoid the "logic scandal" of irrational numbers and keep some relevant conclusions, thus solving the mathematical crisis caused by the appearance of irrational numbers. Eudoxus's solution is realized by directly avoiding irrational numbers with the help of geometric methods. This is a rigid dismemberment of numbers and quantities. Under this solution, the use of irrational numbers is allowed and legal only in geometry, but illegal and illogical in algebra. Or irrational numbers are just regarded as simple symbols attached to geometric quantities, not real numbers. Until18th century, mathematicians proved that basic constants such as pi were irrational numbers, and more and more people supported the existence of irrational numbers. /kloc-In the second half of the 9th century, after establishing the real number theory in the present sense, the essence of irrational numbers was thoroughly understood, and irrational numbers really took root in the mathematics garden. The establishment of the legal status of irrational numbers in mathematics, on the one hand, expands human understanding of logarithms from rational numbers to real numbers, on the other hand, truly and completely solves the first mathematical crisis.
Becker Paradox and the Second Mathematical Crisis
The second mathematical crisis stems from the use of calculus tools. With the improvement of people's understanding of scientific theory and practice, calculus, a sharp mathematical tool, was discovered independently by Newton and Leibniz almost simultaneously in the seventeenth century. As soon as this tool came out, it showed its extraordinary power. After using this tool, many difficult problems have become easy. But Newton and Leibniz's calculus theory is not strict. Their theories are all based on infinitesimal analysis, but their understanding and application of the basic concept of infinitesimal is confusing. Therefore, calculus has been opposed and attacked by some people since its birth. Among them, the most violent attack was British Archbishop Becquerel.
Bishop Becquerel
1734, Becker published a book with a long title "The Analyst; Or give a paper to an atheist mathematician to examine whether the objects, principles and conclusions of modern analytical science are more clear or obvious than the mysteries of religion and the main points of belief. In this book, Becker attacked Newton's theory. For example, he accused Newton that in order to calculate the derivative of, for example, x2, he took x as an increment δX that is not 0, then got (X+δX)2-x2, then divided it by δX to get 2x+δX, and finally suddenly took δX as 0, and got a derivative of 2x. This is "relying on double mistakes to get unscientific but correct results." "Because in Newton's theory, infinitesimal is said to be zero for a while and not zero for a while. Therefore, Becquerel ridiculed infinitesimal as "the ghost of death". Although Becker's attack came from the purpose of maintaining theology, it did seize the defects in Newton's theory and hit the nail on the head.
In the history of mathematics, the Becquerel problem is called "Becquerel Paradox". Generally speaking, Becker's paradox can be expressed as "whether infinitesimal is zero": for the practical application of infinitesimal at that time, it must be both zero and non-zero. But as far as formal logic is concerned, this is undoubtedly a contradiction. This problem caused some confusion in the field of mathematics at that time, which led to the second mathematical crisis.
Newton and Leibniz
Both Newton and Leibniz tried to solve Becker's attack by perfecting their own theories, but they were not completely successful. This puts mathematicians in an awkward position. On the one hand, calculus has achieved great success in application; On the other hand, it has its own logical contradiction, that is, Becquerel paradox. In this case, what is the choice of calculus?
"Go ahead, go ahead, and you will gain faith!" D'Alembert sounded the clarion call to forge ahead. Encouraged by this clarion call, mathematicians in the18th century began to rely more on intuition to create a new field of mathematics, regardless of the imprecise foundation and argument. So a series of new methods, new conclusions and new branches appeared. After a long journey of more than a century, with the efforts of several algebras such as D'Alembert, Lagrange, Bernoulli family, Laplace and Euler, an amazing number of virgin lands have been reclaimed, and calculus theory has been enriched unprecedentedly. 18th century is sometimes even called "the century of analysis". But at the same time, the rough and imprecise work in the eighteenth century also led to more and more fallacies, and the discordant noise began to shake the mathematicians' nerves. Let's take an infinite series for example.
The infinite series S =1-1+1-1+1... What is it?
At that time, people thought that on the one hand, S = (1-1)+(1-1)+... = 0; On the other hand, s =1+(1-kloc-0/)+(1-kloc-0/)+... =1,so isn't it 0 = 1? This contradiction puzzled mathematicians like Fourier, and even Euler, who was later called a mathematician hero, made unforgivable mistakes here. He got it.
1 + x + x2 + x3 +.....= 1/( 1- x)
Then let x =- 1, and you get
s = 1- 1+ 1- 1+ 1………= 1/2！
From this example, it is not difficult to see the chaotic situation of mathematics at that time. The seriousness of the problem lies in that any details in the analysis at that time, such as series, convergence of integrals, arrangement of differential integrals, use of higher-order differential and existence of solutions of differential equations, were almost ignored. Especially at the beginning of19th century, Fourier theory directly exposed the basic problems of mathematical logic. In this way, it is urgent for mathematicians to eliminate disharmony and re-establish analysis based on logic. In the19th century, the necessary period of criticism, systematization and rigorous argumentation has arrived.
Cauchy
The first step to make the analytical foundation rigorous was taken by the famous French mathematician Cauchy. Cauchy began to publish several epoch-making works and papers in 182 1. A series of strict definitions of basic concepts of analysis are given. For example, he began to describe the limit with inequalities, and turned the infinite operation into a series of inequalities. This is the so-called "arithmeticization" of the concept of limit. Later, the German mathematician Wilstrass gave a more perfect "ε-δ" method, which we are currently using. In addition, with Cauchy's efforts, the concepts of continuity, derivative, differential, integral and sum of infinite series are also established on a solid foundation. But at that time, Cauchy's limit theory could not be perfected because the strict real number theory was not established.
After Cauchy, Wilstrass, Dedeking and Cantor, after their independent and in-depth research, they all reduced the analysis basis to real number theory, and established their own complete real number system in the 1970s. Wilstrass's theory can be summed up as the limit existence principle of increasing bounded sequence; Dai Dejin established the famous capital of Germany and Germany; Cantor proposed to define irrational numbers by the "basic sequence" of rational numbers. 1892, another mathematician initiated the "interval set principle" to establish the real number theory. Thus, along the road opened by Cauchy, strict limit theory and real number theory are established, and the logical basis of analysis is completed. The non-contradiction in mathematical analysis is attributed to the non-contradiction in real number theory, so that calculus, an unprecedented grand building in the history of human mathematics, is built on a solid and reliable basis. Rebuilding the foundation of calculus is an important and arduous task, which has been successfully completed through the efforts of many outstanding scholars. The establishment of a solid foundation of calculus ended the temporary confusion in mathematics and declared the complete solution of the second mathematical crisis.
Russell Paradox and the Third Mathematical Crisis
/kloc-In the second half of the 9th century, Cantor founded the famous set theory, which was severely criticized by many people when it was first produced. But soon this groundbreaking achievement was accepted by mathematicians and won wide and high praise. Mathematicians found that starting from natural numbers and Cantor's set theory, the whole mathematical building could be established. Therefore, set theory has become the cornerstone of modern mathematics. The discovery that "all mathematical achievements can be based on set theory" intoxicated mathematicians. 1900, at the international congress of mathematicians, poincare, a famous French mathematician, declared cheerfully: "… with the help of the concept of set theory, we can build the whole mathematical building … today, we can say that we have reached absolute strictness …"
A poet and lead singer
However, the good times did not last long. 1903, a shocking news came out: set theory is flawed! This is the famous Russell paradox put forward by British mathematician Russell.
Russell built a set S: S is made up of all elements that don't belong to him. Then Russell asked: Does S belong to S? According to law of excluded middle, an element belongs to a set or not. Therefore, for a given set, it is meaningful to ask whether it belongs to itself. But this seemingly reasonable question, the answer will be in a dilemma. If s belongs to s, according to the definition of s, s does not belong to s; On the other hand, if S does not belong to S, then S also belongs to S by definition. It is contradictory in any case.
Russell
In fact, this paradox was discovered in the set theory before Russell. For example, in 1897, Burali and Folthy put forward the paradox of maximum ordinal number. 1899, Cantor himself discovered the paradox of maximum cardinality. However, because these two paradoxes involve many complicated theories in the set, they have only produced small ripples in the field of mathematics and failed to attract much attention. Russell paradox is different. Very simple and easy to understand, only involving the most basic things in set theory. So Russell's paradox caused a great shock in mathematics and logic at that time when it was put forward. For example, after receiving a letter from Russell introducing this paradox, G Frege said sadly, "The most unpleasant thing that a scientist encounters is that his foundation collapses at the end of his work. A letter from Mr. Russell put me in this position. " Dai Dejin therefore postponed the second edition of his article "What is the Nature and Function of Numbers". It can be said that this paradox is like throwing a boulder on the calm water of mathematics, which caused great repercussions and led to the third mathematical crisis.
After the crisis, mathematicians put forward their own solutions. I hope to reform Cantor's set theory and eliminate the paradox by limiting the definition of set, which requires the establishment of new principles. "These principles must be narrow enough to ensure that all contradictions are eliminated; On the other hand, it must be broad enough so that all valuable contents in Cantor's set theory can be preserved. " 1908, Tzemero put forward the first axiomatic set theory system according to his own principles, which was later improved by other mathematicians and called ZF system. This axiomatic set theory system makes up for the defects of Cantor's naive set theory to a great extent. Besides ZF system, there are many axiomatic systems in set theory, such as NBG system proposed by Neumann et al. The establishment of axiomatic set system successfully eliminated the paradox in set theory, thus successfully solving the third mathematical crisis. On the other hand, Russell's paradox has a far-reaching influence on mathematics. It puts the basic problems of mathematics in front of mathematicians for the first time with the most urgent needs, and guides mathematicians to study the basic problems of mathematics. The further development of this aspect has profoundly affected the whole mathematics. For example, the debate on the basis of mathematics has formed three famous schools of mathematics in the history of modern mathematics, and the work of each school has promoted the great development of mathematics.
The above briefly introduces the mathematical crisis and experience caused by three mathematical paradoxes in the history of mathematics, from which we can easily see that mathematical paradoxes have greatly promoted the development of mathematics. Some people say that "asking questions is half the solution", and the mathematical paradox is exactly what mathematicians can't avoid. It said to the mathematician, "solve me, or I will swallow your system!" " As Hilbert pointed out in On Infinity: "It must be admitted that in the face of these paradoxes, the current situation we are in cannot be tolerated for a long time. People imagine that in mathematics, a model called reliability and truth value, the conceptual structure and reasoning methods that everyone has learned, taught and applied will lead to unreasonable results. If even mathematical thinking fails, where should we look for reliability and authenticity? "The emergence of paradox forces mathematicians to devote their greatest enthusiasm to solving it. In the process of solving the paradox, various theories came into being: the first mathematical crisis led to the birth of axiomatic geometry and logic; The second mathematical crisis promoted the perfection of the basic theory of analysis and the establishment of set theory; The third mathematical crisis promoted the development of mathematical logic and the emergence of a number of modern mathematics. Mathematics has developed vigorously from this, which may be the significance of mathematical paradox.
Paradox list
1. Barber Paradox (Russell Paradox): Only one person in a village has a haircut, and everyone in the village needs a haircut. Barbers stipulate that only those who can't cut their own hair should be cut. Q: Does the barber cut his own hair?
If the barber cuts his own hair, it violates his agreement; If the barber doesn't cut his hair, then according to his rules, he should cut his hair again. In this way, the barber is in a dilemma.
2. Zeno Paradox-Achilles and Turtle: In the 5th century BC, Zhi Nuo used his knowledge of infinity, continuity and partial sum to trigger the following famous paradox: he proposed that Achilles and Turtle should have a race, and the tortoise should start 65,438+0,000m ahead of Achilles. Suppose Achilles can run faster than the tortoise 10 times. At the beginning of the race, when Achilles ran 1000 meters, the tortoise was still in front of him. When Achilles finished the next 100 meters, the tortoise was still ahead of him 10 meters ... so Achilles could never catch up with the tortoise.
3. The liar paradox: In the 6th century BC, epimenides, a philosopher in Crete, ancient Greece, asserted that "everything Crete said is a lie."
If this sentence is true, that is to say, Immenendez, a Crete, told the truth, but this is contrary to his truth-everything that all Cretes say is a lie; If this sentence is not true, that is to say, Epimenendez, a Crete, lied, then the truth should be: everything that all Cretes say is true, and the opposite is true.
So it's hard to justify it. This is the famous liar paradox.
In the 4th century BC, the Greek philosopher put forward another paradox: "What I am saying now is false." Ditto, this is hard to justify again!
The liar paradox still puzzles mathematicians and logicians. The liar paradox takes many forms. I predicted: "You are going to say' no' next, right? Answer with' yes' or' no'. "
Another example is "My next sentence is wrong (right) and my last sentence is right (wrong)".
4. The paradox related to infinity:
{1,2,3,4,5, ...} is natural number set:
{1,4, 9,16, 25, ...} is a set of numbers squared by natural numbers.
These two sets of figures can easily form a one-to-one correspondence. So, are there as many elements in each set?
5. Galileo Paradox: We all know that the whole is greater than the parts. From the point on line BC to vertex A, each line will intersect with line DE (point D is on AB and point E is on AC), so it can be concluded that DE is as long as BC and contradicts the graph. Why?
6. Paradox of unexpected exam: A teacher announced that there will be an exam in the next five days (Monday to Friday), but he told the class: "You can't know what day it is today, and you won't be informed of the exam until one o'clock in the afternoon to eight o'clock in the morning."
Can you tell me why I can't pass the exam?
7. Elevator Paradox: In a skyscraper, there is a computer-controlled elevator that stops at every floor at the same time. However, Mr. Wang, whose office is near the top floor, said, "Whenever I want to go downstairs, I have to wait for a long time. The stopped elevator always goes upstairs and seldom goes downstairs. How strange! " Miss Li is also very dissatisfied with the elevator. She works in an office near the ground floor and goes to the restaurant on the top floor for lunch every day. She said: "whenever I want to go upstairs, the parked elevator always goes downstairs, and few of them go upstairs." Really annoying! "
What the hell is going on here? The elevator obviously stays on every floor for the same time, but why does it make people close to the top and bottom impatient?
8. Coin Paradox: Two coins are placed flat together, and the upper coin rotates half a turn around the lower coin. As a result, the position of the pattern in the coin is the same as at the beginning; But according to common sense, the coin pattern that turns around the circle for half a circle should be downward! Can you explain why?
9. Paradox of grain heap: Obviously, 1 millet is not a heap;
If 1 millet is not a pile, then 2 millet is not a pile;
If two grains of rice are not piles, then three grains of rice are not piles;
……
If 99999 millet is not a heap, then 100000 millet is not a heap;
……
10. Pagoda paradox: If a brick is taken out of a brick tower, it will not collapse; Draw two bricks, which will not collapse; ..... When the nth brick was pulled out, the tower collapsed. Now start drawing bricks in another place. Different from the first time, when I drew the m-th brick, the tower collapsed In another place, when the tower collapsed, l bricks were missing. By analogy, the number of bricks lost when the tower collapses varies from place to place. So how many brick towers will collapse?
I'm exhausted! !
I hope I can help you ~ ~
Happy new year! !