## Contents

- Expanding the mathematical universe
- Order, relationships and maps
- Regular numbers
- Numbers are rational numbers
- Real number
- Square meters
- Complex numbers
- Quaternions and Octonions
- Groups
- Bell and field
- Vector spaces and relational algebras
- Within product area
- Other words
- Elections
- Additional information
- Divorce
- Strong separatists
- Many parts
- Euclidean structure
- Consolidation
- Sum squared
- Other words
- Elections
- Additional information
- Restoration Act
- Quadratic fields
- Multiple functions
- Diophantine equation
- Other words
- Elections
- Additional information
- Continuity clauses and their use
- Continued section on algorithms
- Diophantine Equation
- Parts in progress
- Quadratic Diophantine equation
- Modular group
- Non-Euclidean geometry
- Completed
- Other words
- Elections
- Additional information
- Hadamard’s problem
- What is the decision?
- Hadamard matrices
- Measurement techniques
- Some Thoughts on the Matrix
- Application of the Hadamard problem
- Graphics
- Groups and codes
- More details
- Elections
- Hensel’s p-adic numbers
- Value fields
- Balance
- Finish
- Invalid fields
- Common area value
- Other words
- Elections

## Preface

There are two types of undergraduate programs in mathematics. On the one hand, there are topics in the curriculum that are assumed to be familiar to every mathematics student, such as linear algebra or real analysis. On the other hand, there are also courses designed to prepare for research given by teachers in their own fields. I think there are many reasons why students need more than this.

First of all, it is important to have common sense and appreciate the work of others, although today the abundance of numbers does not enable anyone to know more than a fraction in depth. In fact, it is sometimes surprising that the relationship between different branches of mathematics forms the basis of many activities of mathematics and the motivation for further development. Second, different branches of mathematics are interesting in different ways and require different skills. It is unlikely that all students studying at the same university have the same interests and abilities as their teachers. Instead, they will only discover their interests and talents by meeting a wider range of people. Third, many mathematics students will not become professional mathematicians but scientists, engineers, or teachers. It is useful for them to understand the nature and order of numbers, and among the benefits of numbers, there must be a group of people in society who have this understanding.

This book attempts to provide such an explanation of the shape and size of the figures. The nexus is number theory, which was originally one of the most obscure and irrelevant branches of mathematics. But we can get a broader picture by looking for more connections with other departments. The topics chosen are not trivial and require effort from the reader. As Euclid said, there is no such thing as the royal road. I generally focus on highly successful results covering a wide range of areas. When I was accused of rolling my eyes at some topics, I said “But what beautiful eyes!” I have no defense other than to say.

The book is divided into two parts. Part A, covering basic mathematics, must be completed in the first year of the first cycle. In order to provide a basis for the following study, Part I discusses the elements and basic structures of the various figures. However, the reader can skip to this section and return later if necessary. Part V, on Hadamard’s deterministic problem, shows that fundamental numbers can have unexpected applications.

Part B is more advanced and is designed to provide the student with an understanding of contemporary mathematics standards. The departments in this section are largely independent, with the exception of the Department.

Although much of the content of the book is common to any introductory work in mathematics, I would like to draw attention here to the discussion of quadratic and elliptic equations. These are special cases of algebraic numbers and algebraic lines, and one may wonder why attention should be restricted to these special cases when the general cases are already well understood and can be developed. My results are as follows. First, full treatment of common cases requires a time commitment that most people cannot afford. Second, these special cases are the most common and constructive methods exist for them rather than general problems. There is another reason. Sometimes in general mathematics things are so simple and broad that a particular case makes sense as an example of a general case. But it is difficult to generalize on the points mentioned, and to meseems more like an improvement on a particular case.

At the end of each chapter of the book I have included a list of selected items that will help the reader make the long journey in their choice. Since there are so many books, each selection must be random, but I hope mine will be interesting and useful.

The computer revolution has made computations possible at the scale and speed of the last century. One effect is a huge increase in ‘search statistics’: the search for patterns. On the other hand, this book is devoted to the ‘mathematics of observation’, that is, the description of nature. I don’t want to hide the fact that the first usually comes before the second. I do not want to hide the fact that some of the answers here were shown after many years of study by great geniuses of the past, and their proofs were improved and simplified by other mathematicians. But once available, good evidence organizes and explains large amounts of statistical data. Often provides ideas for improvement.

This book can certainly be considered a ‘repository of evidence’. We focus on this aspect of mathematics because it is not a special feature of the course, but also because we think its presentation is better suited to a book rather than a tablet or computer screen. According to this method, the evidence itself is carefully selected, and I think very little of it will be of interest even to non-students. Evidence based on general principles was given greater weight than evidence that did not provide specific insight.

Mathematics is part of the culture and activity that people can enjoy. It does not belong to any national, political or religious group, and attempts to do so are destructive. There is a lot of pressure to work harder these days. However, employees of some universities are also evaluated.

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