By yobinbin
01
Galois' last letter
Twenty-decahedrons cannot continue to decompose and cannot be interchangeable in sequence. Therefore, in the case of five equations, it is impossible to find the same combination in symmetric group s5 simply by adding and subtracting and multiplying the five roots. If the five roots are expressed by the multiplier of the medium factor of the equation, then the original coefficient can be generated by repetition, reduction and multiplication. Since it was not possible to introduce the original coefficient, it was suggested that the multiplier alone could not represent a root formula。
This is illustrated by the following quotation from garova. In his letter to his friend chevalier on the night before the duel, he wrote the following words (abel / garois elliptical function)
The argument about the circumstances under which multiplication can be used to solve the equation has been clearly studied. If each of these groups has the same ranking of prime numbers, then the equation can be solved by multiplying. Otherwise, multiplication alone is incomprehensible。
The term “these groups” referred to by garoowa refers to the formations that arise when the groups rotated from their roots. For example, in three equations, symmetric group s3 consists of times and toons. In this case, “these groups” means
And {1, tow} with the "sorting numbers" of 3 and 2 respectively。
In the four equations, symmetric group s4 consists of toto1, toto2, toto3 and times. In this case, "these groups" means {1, Λ1, }2, }]
And {1, ton3} with a score of 2, 2, 3 and 2. Garowa's determination was that because all numbers were prime, equations could be solved by multiplying them。
Proves that the group is similar when the arranged number is p
The circle. At this point, only the p side that requires additions and subtractions can find the same combination of root in the group. Moreover, the original root can be obtained by calculating the p side root. When the root rotation is broken down into these groups, the formula is broken with a multiplier。
However, in the case of five equations, the "these groups" of symmetric group s5 refer to the positive twenties i and {1, ton}. I contain 60 rotations, and 60 is not a prime number. Since the “arrangement number” is not a prime number, the five equations are insoluble according to the determination conditions set out in the last letter from garova。
02
The difficulty of an equation and the beauty of a graphic
The subjects considered above are the next equation in its general form, but the galois approach applies equally to special forms of equation. For example:
Five equations, but five integer root x = 1, 2, 3, 4 and 5. And, n, the equation
And the root of all natural numbers n can be expressed by a multiplier of natural numbers. In expressing the nature of such equations, symmetrical clusters are usually not used, but rather general galois。
While it is not possible to explain in detail what is called the galois, it is a fixed group that applies to each equation. The normal form of n is the n symmetrical group, although in particular forms of equation, the galois group sometimes shrinks。
Galois represents the solution of the equation. For example, because of an equation
There is only one root, so the only root in rotation can only become itself. The galois group at this time is. One equation is simple, so the galois counterpart is simple. In the normal secondary equation
Central, galois is s2 = {1, help}. Because there is a rotation of effort at this time, root cannot be used to square root by a plus and b minus minus。
The higher the dimensions of the equation, the bigger the galois. In the normal five equations, the galois group is s5, and includes a positive twenty-decahedron group, so five equations cannot be multiplied。
However, in the particular form of equations, there are cases of a smaller galois population. Like the five equations that we've put up before
The galois group is the same as a single equation, i. E. . And..
The galois are embedded in the equation
Medium. As long as using the dimensions of the equation equals the form
The prime numbers are proven. Because right now
Repeats the root of the equation for additions and subtractions, as well as square calculations, and can be expressed by the coefficients in the equation (in the case just now 1 and -1). All the roots of the equation can be expressed in square root if squared. If you mark the equation inside the goss plane
That's right
The tip of the side. Since it's square root, it can be mapped, so it's a positive triangle, a positive pentagon, a positive 17-percenter, a positive 257-percenter, a positive 65,557-percenter。
Conversely, the equation for normal natural number n
, which contains not only {1, ton} but also a group for any prime number p
Embedded composition. While the equation is insoluble by square root alone, it can be solved using a normal multiplier。
The concept of numbers needs to be expanded when solving the equation. If it is a single equation of the integer coefficient, the fractions can be solved. The square root of an integer is required for the solver of a secondary equation, and the square root of an integer is required for the solver of three equations. Moreover, in equations with more than five dimensions, there are numbers that cannot be expressed in root. The root of the normal five equations, although not expressed in root, can be expressed in elliptical mode functions。
The garowa group explained to us the number that needs to be used to understand the equation. Garois not only essentially answered “the difficulty of five equations”, but also explained “the difficulty of what is an equation”。
The concept of “group” proposed by garois is widely applied in various fields of mathematics. In sections 1 and 2, we explained the symmetry of the positive triangle by using the concept of clustering, which was derived from the post-galova era. Moreover, the positive-twenty hexadecahedron that appears in section 6 represents the symmetry of geometry. In my eyes, the stereo is more beautiful than the positive polygon in the plane, perhaps because of the complexity of the groups that represent symmetrical clusters. In this case, it can be said that the complexity of the clusters represents the beauty of the graphics。
In 2003, the russian mathematician grigori perelman proved the “pongalé conjecture” to be a worldwide event. There is a link between the pongalé conjecture and the "standard for graphical difficulty". At the beginning of the twentieth century, french mathematician henry pongarai tried to apply the concept of the galowar cluster to geometry. So he proposed a group called the “basic group” to show the complexity of the space in all shapes。
Pongalai believes that there is only one space in three dimensions, the simplest of the basic group. But he did not prove successful. In a situation where space dimension is two-dimensional, this assumption has been widely accepted as correct since time immemorial. In a five-dimensional situation, steven smail successfully certified and won the fields award in 1961。
In four-dimensional cases, michael friedman successfully certified and won the fields award in 1982. Perelman proved the conjecture in the last three dimensions (a certificate every 21 years, which should be a coincidence)。
In fact, perellman won the fields award in 2006 at the same time, although john bauer, then president of the international mathematics league, personally went to st. Petersburg to convince him to accept the award, but in the end it was rejected。
The concept of the “group” that has evolved in the field of mathematics since garowa has been applied in all fields of science since the twentieth century. Einstein, for example, created narrow relativism and broad relativity based on the principle that physical law must be symmetrical. In the field of chemical and material sciences, scientists use the concept of clusters to distinguish molecular and crystallized structures。
Moreover, in the basic particle theory that i have studied, the language of the communities is an essential tool for understanding the fundamental particles and their power. In the light of the foregoing, the concept of a “group” emerged from garova's in-depth reflection on “what is the difficulty of the equation”, and it has made a significant contribution to the development of science and technology。
03
A second soul
The aim of the book is to give you a meaningful life in the twenty-first century, including topics that are commonly used in everyday life, such as how to estimate probability or large numbers that help to judge risk, as well as knowledge that is purely of interest, such as the question of whether the root of the equation can be solved。
Some felt that it was not necessary to teach mathematics in everyday life, such as the root formula of the secondary equation, at the stage of compulsory education, and that during the period of general education, the authors had removed the root formula from the curriculum of secondary schools. However, learning “unworkable mathematics” has some meaning. Because learning mathematics reflects a side of language learning。
A group of indigenous people live in the north-east of australia. The words “left” and “right” are not in their language, so it is customary for natives to use the south-west and north-west for positions, for example, to say, “an ant on your north foot”. As a result, they are very sensitive to the south-east and north-west and have a strong sense of direction and are never lost。
The linguistic structure of japanese and english is very different, for example, the expression of english must be accompanied by the dominant language, although the dominant language of japanese may be omitted. For example, in a conversation like “what did you do yesterday?” “go to the movies”, both sentences omitted the main words。
The university of stanford psychological research laboratory recently conducted an experiment in which they arranged for the english mother tongue and japanese mother tongue to watch a video. In the video, the actors break the vase and turn the milk over. When the video is over, they'll ask the viewer, "who broke the vase?" when people in the video deliberately broke the vases, both the english and japanese mother tongues clearly remembered the breakers. However, when the vase is accidentally broken, the japanese mother tongue hardly remembers who broke it. This is because what you see in japanese is often omitted。
Conversely, there are some unique expressions in japanese. For example, there are many words in japanese that say “i” and “you” and there is a great deal of respect and courtesy. Thus, when using japanese, we are accustomed to judging the relationship between each other and then choosing the corresponding expression according to the relationship。
Language choices greatly affect our feelings and thinking about things around us。
After the demise of the ancient roman empire, charles the great reunified europe. It is said that the great charles has a famous saying: “knowing another language means having a second soul”. Our way of thinking is governed by language. Thus, learning about foreign languages often requires learning about new ways of thinking。
Mathematical language has emerged precisely to help us return to basic principles and to grasp the essence of things as correctly as possible. In chapter 6, cartesian's methodology was quoted, “after the problem has been resolved, then it will be combined to test whether it is complete and whether it has been resolved once and for all” and “unforeseen” matters will not be allowed. And for “small issues, from simplicity to complexity, to begin with easy solutions”, and for the absence of ambiguous expressions, “i will never take the truth as if it were not clearly understood”。
Mathematics should be taught not only in practical ways but also in the capacity to think. Elon mask was quoted at the beginning of chapter 2: “in the real sense of innovation, it must be based on the rationale. “any area is the same, first to discover the most basic truth in this field and then to rethink it。
There are, of course, situations that cannot be resolved in this way. Kobayashi has established japan's recent commentary, which has profoundly influenced the way modern japanese think. He wrote in his title, “assured by the beauty of the flower, but not the beauty of the flower.” in other words, beauty is specific, not an abstract concept。
Mathematics have a limited range of subjects, although its limited range of subjects contains an amazing world. Garois, both in his arms, said himself, “there is a difficult equation, and there is no difficulty with the equation”. But his thinking did not stop here, but he tried to express this “difficult” in mathematical terms, thereby creating a “group” language. Finally, the cluster has become the key to opening the door to the new world of mathematics。
Math is a developing language. At the forefront of science, new mathematics is emerging to express the latest scientific knowledge. The mathematicians and physicists at the caffrey united space research institute, while constantly discovering new mathematics, are also working to solve the mystery of the universe。
New language is created to discuss things that are unprecedented and to answer unresolved questions. It is also one of the greatest intellectual activities of humankind. The book focuses on the mathematical fields that humankind has built for thousands of years, from the era of birun and ancient greece in cuba to the golden era of chinese and arab civilization, the scientific revolution from medieval europe to the renaissance period, the japanese mathematics of the edo era, the french great revolution and the recent german university system to modern society. I think that it's important to have a second soul in contact with these human activities。
