Notes on "Mathematics Appreciation and Discovery"

Mathematics appreciation and discovery

Yu editor

Section 1: From Root Number 2 to Barber Paradox.

First, the emergence of the root number 2-the first mathematical crisis

Ten "beauty theorems": the theorem that the root number 2 is an irrational number ranks seventh, followed by "π is a transcendental number", the four-color theorem and a conclusion of the great mathematician Fermat.

Pythagoras (about 580 BC-500 BC)

Euclid, an ancient Greek geometer, proved that the root number 2 is irrational.

Brother Pythagoras: He is the founder of "philosophy" and "mathematics". The former means "intellectual hobby" and the latter means "knowledge that can be learned". The core view of Pythagoras school is that "everything is a number", that is, everything in the universe can be traced back to integers or the ratio of integers. This school discovered Pythagorean theorem, but it also triggered the first mathematical crisis.

Second, whether infinitesimal is zero-the second mathematical crisis

The calculus founded by Newton and Leibniz is based on infinitesimal, but how small is infinitesimal? Newton would take infinitesimal as the denominator and then cut off the infinitesimal number. Is that an infinitesimal zero? If it is zero, it can't be the denominator; if it is not zero, it can't be cut off. This is the imprecision in calculus.

1734 was questioned by George Becker, the pioneer of subjective idealism philosophy, which led to the second mathematical crisis.

This debate lasted until19th century. 100 years later, the limit theory established by the famous French mathematician Cauchy and later by Wilstrass, Dai Dejin and Cantor laid a strict logical foundation for the calculus theory.

Third, Barber Paradox-The Third Mathematical Crisis

/kloc-In the second half of the 20th century, Cantor founded the famous set theory, which eventually made the building of mathematical science rise steadily.

However, Bertrand Russell, a British philosopher, put forward an assertion: Set theory is not absolutely strict, it is flawed. For example, if a barber wants to cut a person's hair, does he want to cut his own hair? If you don't give yourself a haircut, then it is qualified, then you should give yourself a haircut. If you can cut your own hair, you shouldn't cut your own hair if you don't meet the requirements.

From Euclid to Lobachevsky in the second quarter

1. Euclid and Elements of Geometry

The Elements of Geometry is Euclid's work in the 3rd century BC, and it is known as the "Mathematical Bible".

Three early mathematicians of Alexandria in ancient Greece: Euclid, apollonius and Archimedes.

Gauss (19th century) is recognized as the "king of mathematicians" after Newton. Found the existence of non-Euclidean geometry (which can prove the parallel axiom), but did not put forward it.

Second, Lobachevsky and non-Euclidean geometry

It is generally believed that the founders of non-Euclidean geometry are Lobachevsky and Bolyai.

The essential difference between Roche geometry and Euclidean geometry lies in their different parallel axioms.

Flo geometry also includes Riemannian geometry.

Riemann is Gauss's close disciple.

Klein, a German scientist, gave a unified explanation of non-Euclidean geometry: Euclidean geometry is called "parabolic geometry", Roche geometry is called "hyperbolic geometry" (the sum of internal angles of a triangle is less than 180 degrees), and Riemann geometry is called "elliptic geometry" (the sum of internal angles of a triangle is greater than 180 degrees).

Kant's idealism.

The discovery history of non-Euclidean geometry is also the struggle history of materialism and idealism in geometry.

Section III From Pythagorean Theorem to Fermat's Conjecture

Pythagorean theorem, also known as Pythagorean theorem, is called "the pearl of geometry" and "the eternal first theorem".

Pythagoras proved Pythagoras theorem.

China proved the Pythagorean theorem in Shang Gao in the Western Zhou Dynasty, which was more than 500 years earlier than that in the West.

I. Proof of Pythagorean Theorem

Zhao Shuang's method:

The Highest Prize in Mathematics-Fields Prize

Second, the Algebraic Study of Pythagorean Theorem

In the unified expression of Pythagorean number, the following formula is generally adopted:

17th century, Fermat conjecture:

X n+y n = z n equation, when n >; 2. A set of positive integer solutions cannot be found.

Euler proved that there is no positive integer solution when n=4 and 3.

The British mathematician andrew wiles finally proved Fermat's conjecture in 1995.

Section 4 From Zhouyi Eight Diagrams to Binary Numbers

Leibniz is an encyclopedic scholar who invented calculus and binary system.

There is a cloud in Zhouyi: Tai Chi gives birth to two instruments, two instruments give birth to four images, and four images give birth to gossip. Kun, Gen, Kan, Xun, Zhen, Li, Hui and Gan. The "off state" is indicated by 0.

Section 1 Perfect Numbers and Affinity Numbers

1903, Cauchy published an academic report, 2 67- 1 =193707721x 761838257287, because he denied "267-/kloc-0.

2 P- 1 (P is a prime number) is called mersenne prime in number theory. Most of the big prime numbers found by people belong to mersenne prime.

Prime numbers are also called prime numbers.

First of all, the perfect number

A number is equal to the sum of all its factors (excluding itself) and is a perfect number. For example, 6= 1+2+3, so is 28. (Extension: The moon orbits the earth for 28 days. There were six arts in ancient China: ritual, music, shooting, imperial, writing and counting. Qin Shihuang counted six countries, and there were 28 stars in the sky. )

If 2 n- 1 is a prime number, then the natural number 2 (n- 1) x (2 n- 1) must be a perfect number. Euclid proved this proposition and gave that the following is a perfect number. When n = 2,3,5,7.

The ancient Greek mathematician Nicomachus divided natural numbers into three categories: perfect numbers, abundant numbers and deficient numbers: natural numbers equal to the sum of all their true factors are called perfect numbers, natural numbers greater than the sum of all their true factors are called abundant numbers, and natural numbers less than the sum of all their true factors are called deficient numbers.

Even a perfect number corresponds to the essence of mersenne prime.

Second, the affinity number

The sum of all true factors of 220 is 284, and the sum of all true factors of 284 is 220. Pythagoras called these two numbers "the number of relatives" or "the number of friends", that is, any number of two natural numbers is the sum of the true factors of another number, so these two numbers are affinity numbers.

In 1636, Fermat found the second pair of affinity numbers 17296 and 184 16. Two years later, Descartes discovered the third pair of affinity numbers: 9437056 and 9363584.

In 1747, Euler directly listed 6 1 pairs of affinity numbers, although two pairs were wrong.

Later, thousands of relatives were found one after another.

With the birth of electronic computers, it is found that only 42 pairs of natural numbers are below 1 million, and only 1 3 pairs are below1million.

Mersenne prime in the second quarter

There are infinitely many in mersenne prime.

First, the hard exploration in the era of paper-and-pencil calculus

In the era of paper-and-pencil calculus, only 12 mersenne prime were found.

Second, a major breakthrough in the era of machine computing

In 20 18, Internet mersenne prime Search (GIMPS) project published 5 1 mersenne prime (2 82589933- 1), which is the largest prime number ever discovered by human beings.

Third, the conclusion

Searching for mersenne prime is helpful to improve the traditional computer encryption algorithm.

The number of daffodils and capriega in the third quarter

Narcissus: The traditional name is "cubic regression sign" or "idempotent sign"153 =13+5 3+3.

If an n-bit natural number is equal to the sum of the powers of each digit, it is called an n-bit n-degree regression number.

Peach blossom number: 1634 = 1 4+6 4+3 4+4 4.

Some people call it flower count or flower count.

In 1986, Anthony Dilana, a math teacher, proved that N digits can only become regression numbers through 60 digits at most.

Second, Capulet number

Divide a number into two halves (if it is odd, the high-order digit is supplemented by 0), add it up, and then square it, which is exactly the original number. Such numbers are called "Kapoor Park Jung Su Addendum" or "Ray Split Number", and also called "Square Multiplication Number". Such numbers are 2025, 3025, 980 1 and so on.

(x+y)^2 = 100x+y

The minimum Capulet number is 81((8+1) 2 = 81).

The fourth quarter treasures in the corner

First, the most mysterious number 142857

1/7 = 0. 142857 ...

When the number 142857 is multiplied by 6, a digital period quadrant appears.

Second, palindromes

Reading from left to right is exactly the same as reading from left to left.

123456798765432 1 time is called olive number, which is also a complete square number.

Third, the number of self-control.

The mantissa of the square is equal to the number itself, and such a number is called an automorphic number, for example, 25x25=625.

Fourth, the most unlucky number is 13.

In the East, 13 is a lucky number. Thirteen schools of Buddhism were introduced into China, representing the perfection of merit; Potala Palace 13 floor, Tianning Tower 13 floor.

But in western countries, people are more afraid of the number 13. Judas, a disciple of Jesus, betrayed Jesus, and 13 people attended the last dinner. The date of the dinner coincided with 13, which brought suffering and misfortune to Jesus. So the hotel has no 13 floor, and the airport has no 13 boarding gate.

Five, singular idempotent sum

The sum of the powers of the following two groups of numbers is equal:

From the sum of zeroth power to the sum of eighth power, they are all equal, but the phenomenon that the sum of ninth power is equal to two groups of numbers disappears.

1900, mathematician Hilbert put forward 23 famous mathematical unsolved problems, which were called "Hilbert problems".

In 2000, the Clay Institute of Mathematics in the United States put forward "Seven Millennium Mathematical Problems" (with a prize of $654.38+00,000).

Section 1 Aesthetic Mathematical Theorem: Euler Formula and Basel Series

I Euler formula

Second, the Basel series.

Calculate the sum of the reciprocal of the square of all nonzero natural numbers accurately. The result of Euler's argument is named after Euler's hometown, Basel, Switzerland. This argument uses the McLaughlin series.

The extension of Basel series produced Riemann conjecture:

The zeros of this function, except s=-2, -4, -6 ... are all distributed on the straight line with the real part of 1/2 on the complex plane.

The Code of Cosmic Evolution: Golden Section and Fibonacci Sequence

First, the golden section

Starting from Pythagoras, for any given line segment AB, we should find a point C on it and divide it into two lines, so that the ratio of the length of the longer line segment to the length of the whole line is equal to the ratio of the shorter line segment to the longer line segment. The ratio is (/√5- 1)/2, which is about 0.6 18.

Second, the ubiquitous golden section.

The 30 north latitude line runs through four ancient civilizations.

The navel, throat, knees and elbows are the four key parts of human existence.

We feel most comfortable at 22 ~ 24℃, because the product of normal body temperature of 37℃ and 0.6 18 is 22.9℃. ..

Used in architecture: Parthenon, Taj Mahal in India, Notre Dame de Paris, Eiffel Tower in France.

Leonardo da Vinci's Mona Lisa and Vitruvian people.

Five-pointed star and regular pentagon.

Third, Fibonacci sequence

The petals of sunflower flowers are 2 1, 34, 55. These numbers are related to Fibonacci series: 1, 1, 2, 3, 8, 13, 2 1, 34, 55. ....

"Rabbit Reproduction Problem"

Fibonacci series is intrinsically related to the golden section, and two adjacent Fibonacci series gradually approach the golden section ratio with the increase of serial number.

Golden rectangle: the side length of the new square is equal to the sum of the side lengths of the two nearest squares.

The third section is the masterpiece of oriental mathematics: China's remainder theorem.

China's remainder theorem (also called Sun Tzu's theorem) is the basic theorem of number theory, which is as famous as Wilson's theorem, euler theorem's last theorem and Fermat's last theorem, and is also called the four great theorems of number theory.

Derived from Sun Tzu's calculation, that is, an integer divided by 3, the remainder is 2, divided by 5, the remainder is 3, divided by 7, and the remainder is 2. Find this integer. (The answer is 105n+23)

Solution: 70x2+21x 3+15x2-105n.

Qin's "Nine Chapters" gives a general expression, "Seek a skill for great development".

Han Xin's soldiers are the representative of similar problems.

The fourth quarter mathematics Everest: Goldbach conjecture

"The queen of natural science is mathematics, the crown of mathematics is number theory, and Goldbach conjecture is the jewel in the crown."

Prime number (also called prime number): A number that can only be divisible by 1 and itself.

"Any even number greater than 2 can be expressed as the sum of two prime numbers."

German mathematician Goldbach.

Second, the arduous exploration of Goldbach conjecture

"Narrow the encirclement"

1966, after seven years' efforts, Chen Jingrun proved "1+2": every large enough even number is the product of a prime number plus no more than two other prime numbers.

Appendix to this chapter

First, the most beautiful mathematical formula:

Indian mathematical genius Ramanukin discovered:

Second, "Twin Prime Number Conjecture"

"Twin prime conjecture": There are infinitely many prime numbers P, which makes p+2 also a prime number.

Chen Jingrun gave proof.

Three-color or four-color conjecture

American mathematicians spent 1200 hours on the computer, made 1000 billion judgments, and verified the conjecture.

20 16 Jilin mathematical society proved the "four-color theorem", one of the three major mathematical problems in the world, by mathematical methods.

Section 1 Connotation, Significance and Level of Mathematical Problem Solving

Mathematical problem solving runs through the whole process of mathematical learning.

Sum formula of arithmetic sequence:

Section 2 Solving Problems for Discovery

First, pursue diversification of solutions to problems.

Second, the pursuit of problem-solving program optimization

Third, the pursuit of generative thinking

The first section counts the wonderful flowers in the forest-the golden cicada hulling

First, the golden cicada comes out of its shell until it dies.

Two sets of numbers, the sum is equal, and the sum of squares is equal. Erase from left to right at the same time, or erase from right to left, and the properties remain unchanged.

The sum of the powers of two groups of numbers is equal.

Secondly, construct an idempotent sum array.

You can generate a new array from a known array.

The second quarter hail game

I. Kakuguchi Series

If it is even, it becomes m/2; If it is odd, it becomes 3m+ 1. Repeat this operation, the last number is 1 without exception.

Second, "123" series

Write a natural number at will, write the number of even numbers, odd numbers and integers, and repeat this operation to finally get 123.

Iii. "6 174" series

Indian mathematician Capriga found that writing a four-digit number, sorting from largest to smallest to get a number, sorting from smallest to smallest to get another number, subtracting the two numbers (reducing the large number), and repeating this operation, the final number is 6 174.

The third part discusses the observation and experiment in mathematical discovery.

I. Scientific observation and scientific experiment

Second, mathematical observation and mathematical experiment

The first section magic square and idempotent sum problem

Luo Shu

Each small circle in Luoshu can represent a 1, which is written in the form of numbers as follows:

This is a third-order magic square, and the sum of three numbers in each row, column and diagonal in the figure is 15.

An easy method to construct magic squares of arbitrary odd order (ladder method invented by Robert);

First, the rich third-order Rubik's Cube

There is some mysterious inner connection between magic square and idempotent sum array. The sum of squares of the first column and the third column in Figure 6-2 is equal.

A new method for generating idempotent sum arrays:

Second, the interesting fourth-order magic square

A famous fourth-order Rubik's Cube is the Rubik's Cube on the stone tablet of Tai Su Temple;

The sum of diagonal lines in each row and column is 34. Draw a square casually, and the sum of four numbers in four corners is also 34. What's even more amazing is that if you move the row (or column) to the other side, the positive arrangement will still be a Rubik's cube, as shown in Figure 6-4.

Third, the charming N-order Rubik's Cube

The discovery of tetrahedral volume formula in the second quarter

The formula of triangle is

S=( 1/2)xaxh

or

S=( 1/2)xaxbxsinθ

The volume formula of tetrahedron is

V=( 1/3)xSxh

So is there a second expression of tetrahedron similar to the triangle area formula?

Given two adjacent sides A and B of a triangle, the opposite side is C, and the included angle between the two adjacent sides of the triangle is:

cosθ = (a^2+b^2-c^2)/2ab

The third part discusses: induction and analogy in mathematical discovery

Reasoning is regarded as an important element of mathematics core literacy.

Reasoning is generally divided into perceptual reasoning and deductive reasoning. The self-construction process of a lot of mathematical knowledge is often a process of "guessing before proving". "Guess" is reasonable reasoning, which is embodied in induction, analogy and other reasoning methods. "Proof" is deductive reasoning, also called argumentative reasoning.

First, inductive reasoning, analogical reasoning and deductive reasoning

Inductive reasoning is from special to general reasoning, and induction can generally be divided into incomplete induction and complete induction.

Analogical reasoning is from special to special reasoning, also known as "analogy".

Deductive reasoning is the reasoning from general to special.

Second, inductive reasoning in mathematical discovery

Once a mathematical problem is associated with prime numbers, it may become a meaningful research object.

Goldbach conjecture: Any even number greater than 4 can be expressed as the sum of two odd prime numbers.

Basie's Quadrant Theorem: Any natural number can only be represented by the sum of squares of one, two, three or four. (Except 7=4+ 1+ 1+ 1, it should be expressed by the sum of four squares).

Fermat prime number:

Fermat put forward this conjecture in 1640, but it was rejected by Euler in 1732, because it was not valid when n=5.

Thirdly, analogical reasoning in mathematical discovery.

The first section Steiner-Lemos theorem

It is mentioned in Geometrical Elements that the bisectors of the two bottom angles of an isosceles triangle are equal in length.

Lemos put forward the inverse proposition of this proposition: A triangle with two equal bisectors of internal angles is an isosceles triangle.

The first person to answer this question was the Swiss geometer Steiner.

Average number of people in the second quarter

Given any natural number, do the following:

(1) Calculate its square number first;

(2) Divide the square number into two parts to obtain two new numbers;

(3) Add or subtract the divided two numbers.

If the result is a complete square number, the original number is called the leveling number. For example, 49: 49 2 = 2401-> 24+01= 25 = 5 2.

If the number n of the number A is an odd-even number and 10 A-n is also an odd-even number, then 10 A-n is the odd-even number of n ... such as 5 1 and 49.

The third part discusses: generalization and specialization in mathematical discovery.

Pascal's Hexagon Theorem: If a hexagon is inscribed with a quadratic curve (circle, ellipse, hyperbola, parabola), then the intersections of its three pairs of opposite sides are all on the same straight line.

Section 1 A good education needs a good teacher

Four criteria of a good teacher: ideal and belief, moral sentiment, solid knowledge and kindness.

There is an essential difference between thirst for knowledge and thirst for exploration. Curiosity is the inherent demand of learners for knowledge learning, and what they withdraw is the worship of predecessors' accumulated experience; The desire to explore is the inner desire to know the unknown world, and the retreat is the development of the unknown world.

Section 2 Discovery Teaching: Finding the Way of Teaching Mathematical Knowledge

Myth: knowledge determines everything, but we should also pay attention to whether we understand and are good at thinking.

Teaching for discovery: finding a way out in mathematics problem-solving teaching

Ideal number theory, a new branch of mathematics, benefits from the exploration of Fermat's conjecture.

Ginsburg's "Seven Bridges" problem became the source of graph theory, and mersenne prime's research also promoted the revolution of computer technology.

Paradox and solution: (1) Zhi Nuo's infinite paradox; (2) liar paradox; (3) About solving contradictions.

Goldbach's conjecture, known as "the jewel in the crown of mathematics", has not been deduced so far.

Mclaughlin series:

More mathematicians seem to be more interested in Riemann conjecture.

2 is the smallest prime number (also called prime number) and the only even prime number.

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