About "0"
0, can be said to be the earliest human contact number. Our ancestors only knew nothing and existence at first, and none of them was 0, so 0 isn't it? I remember the primary school teacher once said, "Any number minus itself is equal to 0, and 0 means there is no number." This statement is obviously incorrect. As we all know, 0 degrees Celsius on the thermometer indicates the freezing point of water (that is, the temperature of ice-water mixture at standard atmospheric pressure), where 0 is the distinguishing point between solid and liquid water. Moreover, in Chinese characters, 0 means more as zero, such as: 1) fragmentary; A small part. 2) The quantity is not enough for a certain unit ... At this point, we know that "no quantity is 0, but 0 not only means no quantity, but also means the difference between solid and liquid water, and so on."
"Any number divided by 0 is meaningless." This is a "conclusion" about 0 that teachers from primary school to middle school are still talking about. At that time, division (primary school) was to divide a copy into several parts and figure out how many there were in each part. A whole cannot be divided into 0 parts, which is "meaningless". Later, I learned that 0 in a/0 can represent a variable with zero as its limit (the absolute value of a variable is always smaller than an arbitrarily small positive number in the process of change) and should be equal to infinity (the absolute value of a variable is always larger than an arbitrarily large positive number in the process of change). From this, another theorem about 0 is obtained: "A variable whose limit is zero is called infinitesimal".
"Room 203 105 in 2003", although all of them are zeros, they are roughly similar in appearance; They have different meanings. 0 indicator vacancy of 105 and 2003 cannot be deleted. 0 in Room 203 separates "Building (2)" from "House Number". (3) "(that is, Room 8 on the second floor) can be deleted. 0 also means that ...
Einstein once said: "I always think it is absurd to explore the meaning and purpose of a person or all living things." I want to study all the numbers of "existence", so I'd better know the number of "non-existence" first, so as not to become what Einstein called "absurd". As a middle school student, my ability is limited after all, and my understanding of 0 is not thorough enough. In the future, I hope (including action) to find "my new continent" in the "ocean of knowledge".
Mathematics paper 2
Mathematicization of various sciences
What exactly is mathematics? We say that mathematics is a science that studies the relationship between spatial form and quantity in the real world. It is widely used in modern life and production, and is an essential basic tool for studying and studying modern science and technology.
Like other sciences, mathematics has its past, present and future. We know its past in order to understand its present and future. The development of modern mathematics is extremely rapid. In recent 30 years, the new mathematical theory has surpassed the sum of 18 and 19 th century theories. It is estimated that it will take less than 10 years for each "doubling" of future mathematical achievements.
An obvious trend in the development of modern mathematics is that all sciences are going through the process of mathematization.
For example, physics has long been regarded as inseparable from mathematics. In colleges and universities, it is also a well-known fact that students of mathematics department should study general physics and students of physics department should study advanced mathematics.
Another example is chemistry. We should use mathematics to quantitatively study chemical reactions. We should take the concentration and temperature of the substances involved in the reaction as variables, express their changing laws with equations, and study the chemical reaction through the "stable solution" of the equations. Not only basic mathematics should be applied here, but also "frontier" and "developing" mathematics should be applied.
For example, biology should study the periodic movement of heartbeat, blood circulation and pulse. This movement can be expressed by an equation. By finding the "periodic solution" of the equation and studying the appearance and maintenance of this solution, we can grasp the above biological phenomena. This shows that biology has developed from qualitative research to quantitative research in recent years, and it also needs to apply "developing" mathematics. This has made great achievements in biology.
When it comes to demography, it is not enough just to add, subtract, multiply and divide. When we talk about population growth, we often say what the birth rate is and what the death rate is. So the birth rate minus the death rate is the annual population growth rate? No, in fact, people are constantly born, and the number of births is related to the original base. So is death. This situation is called "dynamic" in modern mathematics. It can't be simply treated by addition, subtraction, multiplication and division, but described by complex "differential equations". Study such problems, equations, data, function curves, computers, etc. Both are indispensable. Finally, it can be clear how each family can have only one child, how to have only two children, and so on.
As for water conservancy, we should consider the storm at sea, water pollution and port design. We also use equations to describe these problems, and then input the data into the computer to find out their solutions, and then compare them with the actual observation results to serve the actual situation. Very advanced mathematics is needed here.
When it comes to exams, students often think that exams are used to check students' learning quality. In fact, the examination methods (oral examination, written examination, etc. ) and the quality of the test paper itself is not the same. Modern educational statistics and educational metrology test the examination quality through quantitative indicators such as validity, difficulty, discrimination and reliability. Only qualified exams can effectively test students' learning quality.
As for literature, art and sports, mathematics is essential. We can see from CCTV's literary and art grand prix program that when an actor is graded, it is often "to remove a highest score" and then "to remove a lowest score". Then, the average score of the remaining scores is calculated as the actor's score. Statistically speaking, "the highest score" and "the lowest score" have the lowest credibility, so they are removed.
Mr. Guan, a famous mathematician in China, said: "There are various inventions in mathematics, and I think there are at least three: one is to solve classic problems, which is a great job; First, put forward new concepts, new methods and new theories. In fact, it is this kind of person who has played a greater role in history and is famous in history; Another is to apply the original theory to a brand-new field, which is a great invention from the perspective of application. " This is the third invention. "There are a hundred flowers here, and the prospects for the development of mathematics and other sciences to comprehensive science are infinitely bright."
As Mr. Hua said in May 1959, mathematics has developed by leaps and bounds in the past 100 years. It is no exaggeration to summarize the wide application of mathematics with "the vastness of the universe, the smallness of particles, the speed of rockets, the cleverness of chemical industry, the change of the earth, the mystery of biology, the complexity of daily life, etc." The greater the scope of applied mathematics, all scientific research can solve related problems with mathematics in principle. It can be asserted that there are only departments that can't apply mathematics now, and they will never find areas where mathematics can't be applied in principle.
Mathematics paper III
What is mathematics?
What is mathematics? Some people say, "Isn't mathematics the knowledge of numbers?"
That's not true. Because mathematics not only studies "number" but also "shape", triangles and squares, which are familiar to everyone, are also the objects of mathematical research.
Historically, there have been various views on what mathematics is. Some people say that mathematics is relevance; Some people say that mathematics is logic. "Logic is the youth of mathematics, while mathematics is the prime of logic."
So, what is mathematics?
Engels, the great revolutionary tutor, stood at the theoretical height of dialectical materialism, profoundly analyzed the origin and essence of mathematics and made a series of incisive scientific conclusions. Engels pointed out that "mathematics is a quantitative science" and "the object of pure mathematics is the spatial form and quantitative relationship of the real world". According to Engels' point of view, it is more accurate to say: mathematics-a science that studies the quantitative relationship and spatial form of the real world.
Mathematics can be divided into two categories, one is pure mathematics and the other is applied mathematics.
Pure mathematics, also called basic mathematics, specializes in the internal laws of mathematics itself. The knowledge of algebra, geometry, calculus and probability introduced in the textbooks of primary and secondary schools belongs to pure mathematics. A remarkable feature of pure mathematics is to temporarily put aside the specific content and study the quantitative relationship and spatial form of things in pure form. For example, it doesn't matter whether it is the area of trapezoidal rice fields or the area of trapezoidal mechanical parts. What everyone cares about is the quantitative relationship contained in this geometry.
Applied mathematics is a huge system. Some people say that it is the part of all our knowledge that can be expressed in mathematical language. Applied mathematics is limited to explaining natural phenomena and solving practical problems, and it is a bridge between pure mathematics and science and technology. It is often said that now is the information society, and the "information theory" which specializes in information is an important branch of applied mathematics. Mathematics has three most remarkable characteristics.
High abstraction is one of the remarkable characteristics of mathematics. Mathematical theory has a very abstract form, which is formed through a series of stages, so it greatly exceeds the general abstraction in natural science, and not only the concept is abstract, but also the mathematical method itself is abstract. For example, physicists can prove their theories through experiments, while mathematicians can't prove theorems through experiments, but can only use logical reasoning and calculation. Now, even geometry, which used to be regarded as "intuitive" in mathematics, is developing in the abstract direction. According to the axiomatic thought, there is no need to know geometric figures. It doesn't matter whether they are round or square. Even tables, chairs and beer cups can be used instead of dots, lines and noodles. As long as the relationship of combination, order and reduction is satisfied, and it is compatible, independent and complete, a geometry can be formed.
The rigor of the system is another remarkable feature of mathematics. The correctness of mathematical thinking lies in the rigor of logic. As early as more than 2000 years ago, mathematicians started from several basic conclusions and used the method of logical reasoning to organize rich geometric knowledge into a rigorous and systematic theory, just like a beautiful logical chain, with every link connected into a line. Therefore, mathematics has always been regarded as a "model of precise science".
Widely used is also a remarkable feature of mathematics. The size of the universe, the tiny particles, the speed of rockets, the ingenuity of chemical engineering, the change of the earth, the mystery of biology and the complexity of daily life require mathematics everywhere. In the 20th century, with the emergence of a large number of branches of applied mathematics, mathematics has penetrated into almost all scientific departments. Not only physics, chemistry and other disciplines are still enjoying the fruits of mathematics widely, but even biology, linguistics and history, which rarely used mathematics in the past, are combined with mathematics, forming rich marginal disciplines such as biomathematics, mathematical economics, mathematical psychology, mathematical linguistics and mathematical history.
Mathematicization of various sciences is a major trend in the development of modern science.
History of mathematics development
This book records the development and changes of elementary mathematics in the world. It can be roughly divided into seven items: the appearance of numbers, the origin and development of numbers and symbols, fractions, algebra and equations, geometry, number theory and description of names, which span thousands of years. It can let readers know the glorious history and development of mathematics. This is an interesting encyclopedia reading, which combines history and mathematics.
The emergence of numbers
First of all, the concept of number appeared.
People are born with the concept of "number". Since primitive people, people can distinguish one, two and three, so they have an understanding of logarithm. In order to represent numbers, primitive people created and used an ancient but clumsy and impractical method-the knot number method. The number of objects is represented by tying a knot on the rope. In order to identify the number, an important counting method appears. This method seems clumsy now, but it is a key step for people to understand mathematics from zero to one. From this clumsy step, people also realize that the explanation of mathematics must be as concise and clear as possible. This is the first understanding of mathematics that has influenced human beings since then, and it is also a key step for human beings to understand mathematics.
The Origin and Development of Numbers and Symbols
First of all, the emergence of numbers
Soon, mankind took another big step. With the appearance of words, the most primitive numbers appeared. What is more gratifying is that people integrate their knowledge into the design. They thought of the method of "taking one generation as the smallest", which is the "carry system" in character representation. Among the numerous numbers, there are binary numbers of ancient Babylon and ancient Roman characters, but the Arabic numerals that have been passed down to this day are universal. They told us that simplicity is the best.
Now there are low-order decimal numbers such as "binary number" and "ternary number". Sometimes people think it is too concise, which makes the data too long and inconvenient to write, and the conversion of decimal Arabic numerals is also very troublesome. In fact, people are higher animals and have a strong understanding ability. Since ancient times, ten has been taken as a whole, so it is customary to use decimals. However, not everything has IQ, and it is impossible to clearly distinguish 1- 10, but two numbers can be expressed in obviously opposite ways. As a result, human beings have created "binary numbers", but they are not convenient to write, and are only suitable for computers and some intelligent machines. But it is undeniable that it has created a new form of digital expression.
Second, the emergence of symbols.
Mathematical symbols such as addition, subtraction, multiplication and division are the most familiar symbols for each of us, because we can't do without them not only in mathematics learning, but also in almost every day's daily life. Don't treat them so simply.
Single, it was not until the middle of17th century that it was fully formed.
French mathematician Soso used some symbols in his three arithmetic papers written in 1484, such as D for addition and M for subtraction. These two symbols first appeared in the commercial speed algorithm written by German mathematician Weidemann. He used "+"for excess and "-"for deficiency.
1, plus sign (+) and minus sign (-)
Addition and subtraction symbols "+","-",1489 German mathematician Weidemann first used these two symbols in his works, but they were officially recognized by everyone from the Dutch mathematician Heuck in 15 14. At 15 14, Heck of the Netherlands used "+"for addition and "-"for subtraction for the first time. 1544, German mathematician Steefel formally used "+"and "-"to express addition and subtraction in integer arithmetic, and these two symbols were gradually recognized as real arithmetic symbols and widely used.
2. Multiplication symbol (×, ...)
163 1 year, the British mathematician orcutt proposed the multiplication with "x". British mathematician Oughtred introduced this symbol in Key to Mathematics published in 163 1. It is derived from the addition symbol+,because multiplication is developed from the addition of the same number. Another multiplication symbol, heriott, was invented by mathematicians. Later, Leibniz thought that "×" was easily confused with "x" and suggested that "×" be used to represent the multiplication sign, so "×" was also recognized.
3. Division (÷)
Division and division symbol "∫" was first popular in continental Europe as a negative sign, and Orkut used ":"to indicate division or ratio. Some people also used fractional lines to express proportions, and later some people combined them into "∫". In the works of Swiss mathematician Laha, "6" was officially used as a division symbol. The symbol "⊙" was first used in Varis, England, and was later popularized in England. In addition to the original meaning of fen, the horizontal line in the middle of the symbol "⊙" separates the upper and lower parts, which vividly represents fen.
At this point, the four major combat symbols have been completed, which is far from being widely adopted by various countries.
4. Equal sign (=)
The equal sign "=" was first used in 1540 by Professor Richter of Oxford University. 159 1 year, the French mathematician Veda was widely used in his works and gradually accepted by people.
mark
First, the generation and definition of scores
The earliest number in human history is a natural number (positive integer). When measuring the average in the future, it is often impossible to get accurate integer results, which leads to scores.
An object, a figure and a unit of measurement can all be regarded as the unit "1". Divide the unit "1" into several parts on average, and the number representing such a part or parts is called a fraction. In the fraction, the denominator indicates how many shares the unit "1" is divided into, and the numerator indicates how many shares there are; One of them is called fractional unit.
The numerator and denominator are multiplied or divided by the same number (except 0) at the same time, and the size of the fraction remains the same. This is the basic nature of fractions.
Scores generally include: true scores, false scores, and scores.
The true score is less than 1.
False score is greater than 1 or equal to 1.
Band score is greater than 1, which is the simplest score. A fraction consists of an integer and a real fraction.
note:
① There cannot be 0 in denominator and numerator, otherwise it is meaningless.
(2) The numerator or denominator in a fraction cannot have irrational numbers (such as the square root of 2), otherwise it is not a fraction.
③ Only two prime factors (2 and 5) in the denominator of the simplest fraction can be converted into finite decimals; If the denominator of simplest fraction only contains prime factors other than 2 and 5, it can become a pure cyclic decimal; If the denominator of simplest fraction contains both prime factors of 2 or 5 and prime factors other than 2 and 5, it can be converted into mixed cyclic decimals. (Note: If it is not simplest fraction, it must be transformed into simplest fraction to judge; The simplest fraction with denominator of 2 or 5 can be converted into finite decimal, and the simplest fraction with denominator of other prime numbers can be converted into pure cyclic decimal.
Second, the history and evolution of scores
Fractions have a long history in China, and the original forms of fractions are different from the present ones. Later, a fractional representation system similar to China appeared in India. Later, the Arabs invented the fractional line, and the expression of the score became like this.
In history, fractions are almost as old as natural numbers. As early as the early days of the invention of human culture, scores were introduced and used because of the need of measurement and average score.
There are records of scores and various scoring systems in ancient documents of many nationalities. As early as 2 100 BC, the ancient Babylonians (present-day Iraq) used fractions with a denominator of 60.
Fractions were also used in Egyptian mathematical literature around 1850 BC.
More than 200 years ago, the Swiss mathematician Euler said in his book General Arithmetic that it is impossible to divide a 7-meter-long rope into three equal parts because there is no suitable number to represent it. If we divide it into three equal parts, each part is 3/7 meters. Like 3/7 is a new number, which we call a fraction.
Why is it called a score? The name "fraction" visually represents the characteristics of this number. For example, if a watermelon is shared equally by four people, shouldn't it be divided into four equal parts? From this example, we can see that fraction is the need of measurement and the need of mathematics itself-the need of division operation.
China was the first country to use scores. In the Spring and Autumn Period (770-476 BC), Zuo Zhuan stipulated that the size of the vassal's capital should not exceed one third, one fifth and one ninth of that of Zhou Wenwang. The calendar of Qin Shihuang's time stipulated that the number of days in a year was 365 and a quarter days. This shows that scores appeared very early in China and were used in social production and life.
Nine Chapters Arithmetic is a mathematical monograph written by China 1800 years ago. The first chapter, Square Domain, talks about four algorithms of fractions.
In ancient times, China used scores 1000 years earlier than other countries. So China has a long history and splendid culture.
geometry
I. Formula
1, planar graph
Square: s = a? C=4a
Triangle: s = ah/2 a = 2s/hh = 2s/a.
Parallelogram: s = ah a = s/h h = s/a.
Trapezoid: s = (a+b) h/2h = 2s/(a+b) a = 2s/h-bb = 2s/h-a.
Circle: s = ∏ r? C=2r∏=∏d r=d/2=C/∏/2r？ = S/∏d = C/∈
Semicircle: s = ∏ r? /2 C=∏r+d=5. 14r
Number of vertices+number of faces-number of blocks = 1
2, three-dimensional graphics
Cube: v = a? = s bottom a s table = 6a? S bottom = a? S plane = 4a? Side length = = 12a
Cuboid: V = ABH = S base H S table = 2 (AB+AC+BC) S side = 2 (A+B) H side length = 4 (A+B+H).
Cylinder: v = ∏ r? H S table = 2 ∏ r? +∏r？ H = s bottom (h+2) s side = ∏ r? H S bottom = ∏ r？
Other columns: v = s bottom h
Cone: v = v cylinder /3
Ball: v = 4/3 ∏ r? S table = 4 ∏ r?
Number of vertices+number of faces-number of edges = 2
number theory
I. Overview of number theory
Humans have been dealing with natural numbers since they learned to count. Later, due to the need of practice, the concept of number was further expanded. Natural numbers are called positive integers, while their opposites are called negative integers, and neutral numbers between positive and negative integers are called 0. Together, they are called integers. (Now the concept of natural numbers has changed, including positive integers and 0)
For integers, four operations can be performed: addition, subtraction, multiplication and division, which are called four operations. Among them, addition, subtraction, multiplication and division can be carried out in an integer range without obstacles. That is to say, any two or more integers are added, subtracted and multiplied, and their sum, difference and product are still an integer. However, the division between integers may not be carried out smoothly within the integer range.
In the application and research of integer operation, people are gradually familiar with the characteristics of integers. For example, integers can be divided into two categories-odd and even (usually called odd and even) and so on. Using some basic properties of integers, we can further explore many interesting and complex mathematical laws. It is the charm of these characteristics that has attracted many mathematicians to study and explore continuously throughout the ages.
The subject of number theory begins with the study of integers, so it is called integer theory. Later, the theory of integers was further developed and called number theory. To be exact, number theory is a subject that studies the properties of integers.
Second, the development of number theory
Since ancient times, mathematicians have always attached great importance to the study of integer properties, but until the19th century, these research results were only recorded in arithmetic works of various periods in isolation, that is to say, a complete and unified discipline has not yet been formed.
Since ancient China, many famous mathematical works have discussed the content of number theory, such as finding the greatest common divisor, pythagorean array, integer solutions of some indefinite equations and so on. Abroad, mathematicians in ancient Greece have systematically studied one of the most basic problems in number theory-divisibility, and a series of concepts such as prime number, sum number, divisor and multiple have also been put forward and applied. Mathematicians of past dynasties have also made great contributions to the study of integer properties, and gradually improved the basic theory of number theory.
In the study of the properties of integers, it is found that prime numbers are the basic "materials" that constitute positive integers. In order to study the properties of integers in depth, it is necessary to study the properties of prime numbers. Therefore, some problems about the properties of prime numbers have always been concerned by mathematicians.
By the end of18th century, the scattered knowledge about the properties of integers accumulated by mathematicians in past dynasties was very rich, and the conditions for sorting it out and processing it into a systematic discipline were completely mature. Gauss, a German mathematician, concentrated the achievements of his predecessors and wrote a book called Arithmetic Discussion, which was sent to the French Academy of Sciences in 1800, but the French Academy of Sciences rejected Gauss's masterpiece, so Gauss had to publish it himself in 180 1 year. This book initiated a new era of modern number theory.
In On Arithmetic, Gauss standardized the symbols used to study the properties of integers in the past, systematized and summarized the existing theorems at that time, classified the problems to be studied and the methods of will, and introduced new methods.
With the development of modern computer science and applied mathematics, number theory has been widely used. For example, many research results within the scope of elementary number theory are widely used in calculation methods, algebraic coding, combinatorial theory, etc. It is also reported in the literature that some countries now use the "Sun Tzu Theorem" to measure the distance and calculate the discrete Fourier transform with the original root and the original exponent. In addition, many profound research results of number theory have also been applied in approximate analysis, difference set, rapid transformation and so on. Especially due to the development of computer, it is possible to approximate continuous quantity with the calculation of discrete quantity and achieve the required accuracy.
Third, the classification of number theory
elementary number theory
Refers to the number theory problems that elementary algebra deals with not exceeding the high school level. The main tools include integer division and congruence. Important conclusions are China's remainder theorem, Fermat's last theorem, quadratic reciprocity law and so on.
analytic number theory
With the help of calculus and complex analysis, problems about integers can be mainly divided into two categories: product theory and addend theory. The product number theory discusses the distribution of prime numbers by studying the properties of product generating functions, among which the prime number theorem and Dirichlet theorem are the most famous classical achievements in this field. Additive number theory is the possibility and representation of studying integer additive decomposition, and Waring problem is the most famous topic in this field. In addition, such as screening method and circle method are all important topics of this kind. Chen Jingrun, a mathematician in China, solved Goldbach's conjecture with the screening method in analytic number theory.
algebraic number theory
It is a branch that extends the concept of integer to algebraic integer. In the study of algebraic integers, the main research goal is to solve the problem of indefinite equations more generally, and in order to achieve this goal, this field is particularly closely related to algebraic geometry. The concepts of prime number and divisibility are established.
Digital geometry
It was founded and laid the foundation by German mathematician and physicist Minkowski. This paper mainly studies the distribution of integers (in this case, lattice points) from the geometric point of view. The basic object of geometric number theory research is "spatial grid". In a given rectangular coordinate system, the point whose coordinates are all integers is called the whole point; A set of all points is called a spatial grid. Spatial grid is of great significance to geometry and crystallography. The most famous theorem is Minkowski theorem. Because of the complexity of the problems involved in geometric number theory, it needs a considerable mathematical foundation to study it in depth.
Computational number theory
With the help of computer algorithms, problems of number theory, such as prime number test and factor decomposition, are closely related to cryptography.
Transcendental number theory
It is particularly interesting to study the transcendence of numbers, especially Euler constant and specific Zeta function values.
Combined number theory
Using the skills of combination and probability, some complex conclusions that cannot be handled by elementary methods have been proved to be unconstructive. This is an idea initiated by Edith.
Fourth, the jewel in the crown.
The position of number theory in mathematics is unique. Gauss once said, "Mathematics is the queen of science, and number theory is the crown in mathematics". So mathematicians like to call some unsolved problems in number theory "crown jewels" to encourage people to "choose".
A few "pearls" are simply listed: Fermat's last theorem, twin prime number problem, Goldbach conjecture, corner-valley conjecture, integer problem in a circle and perfect number problem. ...
Verb (abbreviation for verb) The achievements of the people of China.
In modern China, number theory was also one of the earliest branches of mathematics. Since the 1930s, he has made great contributions to analytic number theory, complexity equation and even distribution, and first-class number theory experts such as Hua, Min Sihe and Ke Zhao have emerged. Among them, Professor Hua is most famous for his research on trigonometric sum assignment and heap prime theory. After 1949, the study of number theory has made great progress. Especially in the research of "screening method" and "Goldbach conjecture", it has made outstanding achievements in the world. Especially after Chen Jingrun proved in 1966 that "a big even number can be expressed as the sum of the products of a prime number and no more than two prime numbers", it aroused strong repercussions in the international mathematics community, praising Chen Jingrun's paper as a masterpiece of analytical mathematics and the glorious culmination of screening method. So far, this is still the best result of Goldbach's conjecture.
Name description
Euclid's Elements of Geometry was published around 300 BC.
The author of Zhou Pi Ai Jing is unknown, and the time is earlier than the first century BC.
The author of "Nine Chapters of Arithmetic" is unknown about the first century AD.
The author of Sunzi Suanjing is unknown, Southern and Northern Dynasties.
Geometry Descartes 1637
Mathematical principles of natural philosophy Newton 1687
Introduction to Infinite Analysis Euler 1748
Differential Euler 1755
Integral calculus (three volumes) Euler 1768- 1770
Gauss 180 1 year arithmetic query
The prime number of Hua Duiji is about 1940.
Choose any paragraph! ! !