[Original] What is mathematics?

In the 1940s S, the great mathematician R. corang used a book to tell the story of mathematics. Professional Mathematics seems too difficult and abstract for ordinary people. In today's education system, even under the influence of Western pragmatism, Chinese college students basically use mathematics to cope with the college entrance examination and postgraduate entrance exams. They have almost no interest in mathematics. Higher Mathematics has become one of the subjects with the highest degree of subject failure. However, some mathematicians have pointed out that mathematics is a subject that every human member should be proficient in. Mathematics leads the scientific progress of mankind, and mathematics is centered around all aspects of our work. Engineering cannot do without calculus and algebra knowledge. Both material mechanics and theoretical mechanics are represented in the form of differentiation, and points are used to calculate complex areas and volumes. Differential ry and linear algebra are also used for mechanical analysis and computer graphics. Linear Algebra and probability algorithms are also used for optimization. The operational research of management science is also based on the optimization theory.
Mathematics can be classified by continuous mathematics and discrete mathematics. Continuous mathematics is centered on continuous analysis based on calculus, while discrete mathematics is centered on developing mathematical tools.
Substitute mathematics is the basis and tool of all mathematics. With the development of classical equations, modern mathematics forms the basic group theory, model theory, model theory, and Algebra Structure of modern mathematics. Modern algebra is basically based on group theory. Group theory comes from solutions to algebraic equations. Algebra research prior to the 19th century was basically centered on the solution of the First-dimensional high-order equation. However, after several hundred years of efforts, people found only four ordinary algebraic solutions for the number of equations and below by using the formula method. However, there is no solution to the unary High-Order Equations of five or more times. In the 19th century, the geniuses Abel and galova became interested in the solution of the higher-order equation, after research, they believe that more than five general unary High-Order Equations do not have a root equation as we are familiar with the unary quadratic equation (the essence is that they cannot be used with the outsourcers and are called non-algebraic solutions, the geometric problem is that the ruler method cannot be used to create a graph with more than five sides, if an equation has an algebraic solution, it means that any polygon of the number of edges can be made using the classical method). Of course, it must be noted that general equations do not mean that special equations do not work. At that time, both the mathematicians Gao Si and rangeland realized more than five times that the general one-dimensional high-order equation may not have an algebraic solution. At that time, the young Norwegian mathematician Abel made incomplete proof of this, but did not receive the attention of mathematicians, and died at the age of 27. The young French mathematician galova, who later founded the "group theory", proved the problem that had plagued the world for centuries. At that time, he was only in his twenties and sent his paper to the genius mathematician Gauss, gauss thought that he could not understand it and did not pay much attention to it. At that time, due to the French Revolution, galova was a revolutionary who died in a duel at the age of 21. Galova left his theory to a friend one night before his death with 32 pages of paper. It wasn't until 14 years after his death that his manuscript fell into the hands of the mathematician Liu weier. Liu weier carefully read and compiled galova's manuscript into a group theory. From then on, the mathematicians of the world started a enthusiastic research on the group theory, people found that group theory is a powerful idea and tool that can easily solve many difficult problems. Group theory is the basis of modern mathematics, but it is so profound that the principal of Wuhan University who engaged in calculus publicly said that he himself does not understand group theory. Of course, the development of group theory is not only used to solve the problem of the one-dimensional high-order equation. The group theory originally used to solve higher-order equations can basically be considered as a replacement idea. The solution group (solution space or solution set) of the equations that can be solved by algebra is a replacement group. Each high-order equation corresponds to a solution set or a solution group. The problem of solving this high-order equation is transformed into studying the nature of the group (whether it can be replaced ). Take the quadratic equation as an example. Ax ^ 2 + bx + c = 0. It is interpreted as X1 and X2, and X1 and X2 are considered as two unknowns, as long as the Linear Equations containing two equations are found, X1 and X2 can be obtained. From the knowledge of linear algebra and combination, the linear equations of X1 and X2 are nothing more than X1 + X2 =? X1-x2 =? However, we know that such equations are always solvable. Group theory holds that X1 + X2 and X1 * X2 are replaceable polynomials for addition and multiplication operations, and these two are the most basic replacement polynomials for this equation, while x1-x2 is not replaceable polynomial, but (x1-X2) ^ 2 is replaceable, other polynomial can be obtained by the most basic. We will naturally think of replacing X1 and X2 with the original quadratic equation to obtain a X1 ^ 2 + B X1 + c = 0, A X2 ^ 2 + B X2 + c = 0. Subtract the two equations to remove C, it can be reduced immediately (because the polynomial in the form of x ^ N-y ^ N can be decomposed into the form of (x-y) (a polynomial) by the quadratic theorem, so subtraction can certainly remove the constant and x-y), and get X1 + X2 =-B/A. This is one of the results that we often need to verify after solving the quadratic equation in high school; then, the two equations are added, and the obtained results are substituted into the equation to obtain X1 * X2 = C/. Then find the x1-x2 is easy, (x1-x2) ^ 2 = (X1 + x2) ^ 2-4x1 * X2. This gives us the root-seeking formula that we often use in high school. Why is it so troublesome. Because this method can always find the form of the solution of the general equation, not just the form of the algebraic solution, the formula of the quadratic equation was found by China and the Greek in BC, generally, three equations, four equations, took the skills of smart people in the world for centuries 16 and 17. According to the group theory, it is proved that more than five general equations are not in the form of Polynomial multiplication. When talking about this, we didn't talk about the essential problem of solving equations by group theory. Replacement polynomials use the root symmetry. For specific equations, according to the arrangement and combination, there are only a few replacement polynomials corresponding to the equation, and the most basic one is X1 + X2 +... + Xn and X1 * X2 *... * With the increase of the number of equations, it is more difficult to combine n linear equations. through strict proof, for more than five "General Equations ", it is impossible to combine so many linear equations for root.
Number theory studies the nature of numbers. Number theory studies natural numbers. It can be said that it is the most natural mathematics. A mathematician said that God has created natural numbers, and other numbers are invented by humans. Number theory is regarded as pure mathematics, that is, it is too far away from practical application. However, the number theory results of a few hundred years ago seem to have been widely used in a few hundred years. Modern cryptography is based on number theory. The common key cryptography system is based on a large number of prime factors. The Euclidean era proves that the prime number in natural numbers is infinite. As the number increases, the number of prime numbers becomes fewer and fewer. Compared with the total number of natural numbers, the number of prime numbers tends to be infinitely zero, but the total number is not zero. We can use the reverse verification method to prove that a series of prime numbers are always a prime number, so we can easily construct any size of prime numbers. However, it is difficult to prove that a number is a prime number. It is more difficult to break down a large number as a prime factor, the hundreds of digits use the fastest algorithm and the fastest computer decomposition prime factor for almost many years. Modern number theory also involves the study of irrational numbers. The irrational number is essentially an infinite series. For example, the percentile of the circumference rate can be expressed as a triangular series, but it is not a unique representation. The computing efficiency of different representations varies greatly, in ancient China, the mathematician zu chongzhi calculated thousands of polygons to determine the accuracy of the cosine to 3.1415926-3.1415927, which was the year before the world. The mathematician ramanokin of modern Indian genius has also studied the accuracy of Gini and obtained a strange series representation. The first entry of the series makes the accuracy of Gini reach eight places, which greatly reduces the computational workload. Ramanokin was born in India as a poor genius mathematician. He was not passing formal university education due to his interest and obsession with mathematics. He studied mathematics by reading a theorem manual with few proofs, in his book, he invented thousands of new theorems through inspiration and made notes. He did not understand proofs and seldom understood other books, so that some of his theorems were discovered by some mathematicians earlier, he was a high-yielding mathematician. During his studies in the UK, he found six new theorems on average every day. He was the authoritative mathematician at that time. He discovered his genius and got him to the UK, he is also the mentor of Chinese mathematician Hua Luogeng. He is very respectful of ramanokin. He scores the mathematician's talent, scores ramanokin 100, gives himself 25, and gives him 85 points at that time, he believes that the genius of ramanokin can be even higher than or even higher than that of Euler and yarco. In these years, India has also been glorious because of the world-class genius mathematician ramanokin, but ramanokin has suffered from disease and war, when I was 33 years old, I died and left some examples of the theorem that I used to prove. An American mathematician began to prove the full theorem of ramanokin in 1970s and completed it in the near future, his conclusion is that ramanokin's theorem is basically correct, and only a few have defects. The mathematician won the fielz Award (the Nobel prize in the field of mathematics) for his work ). At one time, the modern number theory conference set off the topic of lamanoujin. The proof of a theorem of lamanoujin is sufficient to publish a paper in an authoritative magazine. The proof of ferma's conjecture took more than 300 years for the wise man to solve the problem in 1994. Ferma conjecture itself is a simple equation. X ^ N + y ^ n = Z ^ n when n is greater than 3, there is no natural number solution. When n = 2, there is an infinite solution to the famous stock theorem. This problem proved to be far more difficult than previously imagined. To prove this theorem, mathematicians have invented many tools, including group theory and model theory. This theorem was finally proved by American mathematician Charles to have more than 100 pages. To understand this theorem.
The center of world mathematics. When I was studying mathematics in China and India, mathematics in Europe was not developed. Later, Greek mathematicians formed a school of thought to systematically study mathematics and gradually develop mathematics into a discipline. The systematic research of mathematics was Euclidean, having written "ry originally", our middle school elementary mathematics is basically within the scope of this book. Several centuries later, with the development of the industrial revolution. The world's mathematical center has been transferred to the UK. The mathematician represented by Newton has developed the idea of calculus and opened up the field of analytical mathematics. Later in the 19th century, the world's mathematical center was transferred to Germany, centered on the University of Göttingen, represented by genius mathematicians such as Gauss, and focused on number theory to develop and study various mathematical branches, it is worth mentioning that Gauss, at the age of 24, laid the foundation for near-world advanced algebra with Arithmetic research. The center of modern mathematics is transferred to the United States. The 20th century marks the beginning of modern mathematics. mathematics has developed into a wide range of branches. However, it is generally based on group theory, such as advanced algebra, Logical Algebra, fanfare theory, graph theory, and composite theory, in addition, the algebraic ry crossover forms the differential ry and topology. With the development of computers, these mathematics have created new vitality in recent decades. History shows that the world's mathematical center is basically transferred with the development of the economy. Generally, the country with the most advanced mathematics is the strongest. Where is the next mathematical center? Hope we are in China.