Figure 1 Academician Zhou Xiangyu
Zhou Xiangyu is a researcher at the Academy of Mathematics and Systems Science of the Chinese Academy of Sciences, an academician of the Chinese Academy of Sciences, and an academician of the Academy of Sciences for Developing Countries. Academician Zhou is mainly engaged in the research of multivariate and complex geometry, and has made a series of internationally leading research results in the field of multivariate, which proves the expansion of the future fluorescent tube conjecture and the Sergeev conjecture, and solves Demailly's strong openness conjecture on the multiplier ideal layer with his collaborators. He has won the National Science Foundation for Outstanding Young Scholars, the Qiushi Outstanding Young Scholars Award, the Chen Provincial Mathematics Award, the Second Prize of the National Natural Science Award, and the Chen Jiageng Science Award.
Academician Zhou talked about the history of the emergence of complex numbers, expounded the magical role of complex numbers and complex functions, explained that "imaginary numbers" are not virtual and the "useless use" of mathematics, and combined with ancient Chinese mathematical thought.
This article is compiled by Academician Zhou Xiangyu on December 16 at the Shahe Campus of Beihang Airlines (partial content), and published with the authorization of the mathematical graticule.
At the beginning of the emphasis on several key words of the complex variable function, among which there are variables, Descartes said that this is an unknown and undecided quantity, and earlier in China we called it "Tianyuan", like Li Ye and others also formed a set of Tianyuan techniques. The four masters of the Song and Yuan Dynasties (Li Ye, Qin Jiushao, Yang Hui, Zhu Shijie) had very important and brilliant deeds in the development of ancient Chinese mathematics, and by the time of Zhu Shijie, he could already make four systems of variable polynomial equations. In fact, it is very important to introduce Tianyuan to mathematics, which can be said to be a turning point in mathematics, because then there are variables. Incidentally, we now refer to "algebra" as "algebra", which was translated by Mr. Li Shanlan, using Veda's point of view, using symbols instead of numbers.
Figure 2 Li Ye
Functions can be seen as connections (relationships) between variables, a very basic concept in mathematics, and a language for describing change and motion. Li Shanlan once gave a definition: the formula contains heaven, which is a function of heaven. This "heaven" is "tianyuan", and the "function of heaven" is the function of variables. Early researchers abroad, such as Orem, Descartes, Fermat, etc., considered the trajectory of the graph to study the function. Functions are a bridge between numbers and shapes. In modern times, it is often mentioned that the image of the function is mentioned, and if there is a function, there is a corresponding image. We often say "point motion forming", the point as a variable is changing, naturally forming a shape. In the past, the research was on the more classical curves, and after the function came out, it was possible to study an infinite number of arbitrary curves. Therefore, functions have the symbolic meaning of algebra, but also have geometric meaning, and can also reflect the laws of physics. For studying the laws of physics, like Stevin, Kepler, and Galileo on mathematical physics, functions are used as an essential tool.
The word "function" was coined by Leibniz, and "variables", "parameters", and "parameter variables" were all introduced by Leibniz. The words he introduced are still in use today, and The "flow" that Newton used is now largely no longer used. In the past, functions were only incidental and not the main object of study, but from Euler onwards, functions were considered to be a major object of study.
With a function naturally studying its limits, a common starting point for convergence, continuum, differentiation, integration, etc. in university courses is the limit. In ancient China, the idea of limits was very rich. The famous huishi has a saying: "A foot of the scepter, the day takes half of it, and the world is inexhaustible" (see Zhuangzi's "Tianxia Chapter"). This is written as a mathematical language like this:
You can't write a finite term, you have to write an infinite item. Another Mozi said: "A line segment is divided into two halves from the midpoint, take half, and then break the half into two halves, still take half of it, until it cannot be divided, it is naturally a point." "This is actually the interval set theorem, which I call mozi semipartition. We can use the Mozi semipartition method to prove the compactness theorem, the polypoint theorem, the finite coverage theorem, and the zero-point existence theorem of continuous functions.
Figure 3 Mozi
"There is poverty" and "infinity" are common terms used by Mojia. Later, Liu Hui invented the "circumcision technique" - "the fineness of cutting, the loss of mili, the cutting and cutting, so that it can not be cut, then it is integrated with the circle, and nothing is lost." This passage itself speaks of a limit. Later, Zu Chongzhi and his son Zu Hui wrote a book called "Embellishment", which has now been lost. "Embellishment" has a continuous meaning, and I think there is a lot of extreme thinking in the book. Among them, there is a very famous ancestral principle: "If the power potential is the same, the product cannot be different." This means that there are two cubes, and if their parallel cross-sectional areas are equal, then their volumes are equal. This is called the Cavalieri principle abroad. Zu Hui raised and answered the remaining question of Liu Hui's request for the ball with a Mou He Square Cover earlier.
Today's engineering students study complex variable functions is very basic and very common. When Qian Xuesen was studying at the California Institute of Technology, he studied the theory of complex variable functions, which was at the forefront at that time. In my opinion, if you don't understand complex functions, you probably can't be called a good engineer.
Here I would like to emphasize that "imaginary numbers" are not false but also reflect a very important concept, that is, "useless use". When the plural was first studied, it was felt that it was not practical, but it is now widely used. In the former Soviet Union, Lavlentiev and Shabat wrote a book, Methods of Complex Variable Function Theory. Two brilliant mathematicians in this book illustrate many examples that reflect the important applications of complex variable functions, including in fluid dynamics, gas dynamics, elastic mechanics, electromagnetism, electrical engineering, circuit calculations, wing design, and so on.
In fact, "useless use" was first proposed by Zhuangzi in "Zhuangzi 'The World on Earth", and he had already reminded us at that time that "everyone knows the useful use, but the unknown useless use is also", and the useless use is indeed very profound. He gave an example. A man saw that a big tree was very dense, and he asked the logger, "Why don't you cut it down?" The logger said that if the tree was used to make a boat, it would sink; if it was used to make doors and windows, it would seep out and not close; if it was used to make a pillar, it would be bitten by insects and not reliable; if it was used to make a coffin, it would decay easily, so it was of little use. As a result, Zhuangzi sighed, you see it is useless, but it lives a long time, it is of great use; in addition, it can also allow many people and livestock to shelter from the rain under it, because it is leafy.
Figure 4 Zhuangzi
It can be seen that Zhuangzi is very thoughtful, and he is a master of dialectics. "Useless use is great use" - useless use is the greatest use. It didn't work at the time, but it worked later; it didn't work from that point of view, but it was of great significance from another point of view. The scientific research of "useless use" lies in the construction of scientific knowledge system. In the "Zhou Yi Zhi Shang", it is said that "exploring the hidden treasures, hooking deep and far away", and "gewu zhizhi", which makes the essence of scientific research very clear. The true meaning of scientific research is to explore esoteric and hidden problems, to explore the unknown to obtain new knowledge, and to build a scientific knowledge system. Xu Guangqi said: "Useless use, the basis of public use", people find many applications through the scientific knowledge system to benefit mankind.
Although imaginary numbers were not practical at the time, they were a major contribution to the mathematical science knowledge system. So in my opinion, this is a use in itself, you can't limit the word "use" to practicality, scientific knowledge system is a precious wealth of mankind, is a priceless treasure. For example, Apollonius, who studied conic curves, had neither a practical background nor a practical purpose at that time, and was completely "useless". Kepler discovered the law of planetary motion, the trajectory of the planet is elliptical; people have found that the trajectory of the shell is a parabola; and the current positioning system is related to hyperbolic curves. Why is the positioning system called a hyperbolic system? The difference in time for the signal to be received by the satellite is always a fixed value for light or signal sent by a ground object, and the speed at which the signal is transmitted is the same, so that the distance between the object and the two satellites is a fixed value. A moving point, if the difference to two fixed points is a fixed value, this is a hyperbolic one. The multipoint positioning system uses this idea to locate, so it is called a hyperbolic system.
Topology in mathematics is also useless. Won the Nobel Prize in Physics in 2016 because of the discovery of topological phases and topological phase transitions, and now topological insulators and topological materials are topologically related research and are very active, while topology was not practical in the past. This year's Nobel Prize in Physics was awarded to Penrose et al. for their work on the discovery of black holes, penrose introduced basic mathematics to study the singularity of space-time, predicting the existence of black holes; his one-hour conference report at the 1978 International Congress of Mathematicians was titled "Complex Geometry of Nature." There is also Boolean study of the laws revealed by human thinking activities (the laws of logical thinking), the discovery of Boolean algebra, when there was neither practical background and practical needs, nor practical purpose, after which a very important practicality was produced, and now plays a very basic role in artificial intelligence. In 1938, Shannon, in his master's thesis, noted the similarity between telephone switching circuits and Boolean algebras, that is, the "true" and "false" of Boolean algebras and the "on" and "off" of circuit systems. So he used Boolean algebra to analyze and optimize switching circuits, laying the theoretical foundation for digital circuits. Boolean algebra is the basis of the chip. Huawei attaches great importance to basic scientific research, and Professor Arikan of Turkey has discovered polarization codes, which greatly improves the performance of 5G coding. In 2016, the International Organization for Standardization of Mobile Communications 3GPP identified a channel coding technology scheme for 5G enhanced mobile broadband scenarios, in which Huawei's polarization code became the coding scheme for the control channel. Arikan's paper is like a mathematical paper. These are excellent examples of mathematical "uselessness".
Here's my understanding of the parable of The Fool Moves the Mountain, in addition to the usual explanations.
A pioneer in public consultation. As the head of the family, Yu Gong did not engage in a single word hall before moving the mountain, "gathering in the room and plotting", "his wife is suspicious", and adopted his wife's reasonable suggestions. This vividly explains the essence of consultation, such as having things to discuss, people's affairs to be discussed, not engaging in formalism, true consultation, consultation before decision-making, and decision-making based on science.
Figure 5 Yugong moves the mountain
In addition, "Yugong Moving Mountains" contains profound mathematical ideas, why? Yu Gong's reply to Zhi Shuo actually contained two very important mathematical principles:
The first half is equivalent to defining natural numbers, recognizing the infinity of natural numbers (known abroad as Pineo's theorem, completed at the end of the 19th century).
The second half is actually the cornerstone of real number theory to measure the correctness of calculus - Archimedes' principle.
"In my death, there are sons who have survived; sons and grandchildren, and grandchildren; sons and sons, sons and grandchildren; children and grandchildren." In fact, here is the definition of natural numbers, recognizing the infinity of natural numbers. In this way, natural numbers are defined, revealing the commutative and union laws of natural numbers.
The generational sets of the Fool's sons and grandsons correspond to the natural number sets, and the generational sets of the Fool's sons and grandsons are also the equivalent sets of the Fools, here, the two descendants of the Fools are called equivalent and only if the two descendants belong to the same generation. If the Duke of Yu himself corresponds to 0, his descendants correspond to 1, and his grandchildren, the children of his sons, correspond to 1+ 1=2; if the descendants of The Duke of Yu correspond to n, the descendants of the descendants correspond to n + 1. The method of defining natural numbers in this way can be called the model of the fool's grandson.
It is easy to see the operation law of natural numbers from the model of the fool. Let's look at the commutative law first: for example, 1+2 and 2+1. In the model of the Foolish Prince, 1+2 corresponds to the grandchildren of his sons, that is, great-grandchildren; while 2+1 corresponds to the children of his grandchildren, that is, great-grandchildren. So 1+2 = 2+1. Let's look at the law of union: for example, in 1+1+1, the first two 1s are added, that is, 2+1; the last two 1s are added to be 1+2, as just explained, so there is (1+1)+1=1+(1+1).
"Children and grandchildren are infinitely poor, and the mountains do not increase, so why not be bitter and not peaceful?" The amount of earth and stone in the mountain is b>0 (which may be very large), and the amount of soil and stone dug by each generation of the Yugong family is a>0 (a may be very small). One generation digs a, two generations are 2a, and in the nth generation, that is na. Because "the mountain does not increase", it is possible to set b as a constant. "The children and grandchildren are infinitely scarce", which means that the natural numbers 1, 2, 3,......,n,...... It can be infinite. So the fool asserts that a natural number n can always be found, making na >b. This is known as archimedes' principle.
The idea of yugong also contains the ultimate meaning of n tending to infinity.