Introduction to mathematical symbols
Mathematics (hanyu pinyin: shùXué；; ; Greek: μ α θ η μ α κ; English: Mathematics/Math) comes from the ancient Greek word μθξμα(máthēma), which means learning, learning and science, and has a narrow and technical meaning-"mathematical research". Even in its etymology, its adjective meaning and learning-related will be used to refer to mathematics. Its plural form in English and as the plural form of mathématiques in French +es can be traced back to the Latin neutral plural (Mathematica), which is Cicero's plural from Greek τ α α θ ι α τ κ? (ta mathē matiká). In ancient China, mathematics was called arithmetic, also called arithmetic, and finally changed to mathematics.
Mathematics is a science that studies concepts such as quantity, structure, change and spatial model with symbolic language. Mathematics, as an expression of human thinking, embodies people's aggressive will, meticulous logical reasoning and pursuit of perfection. Although different traditional schools can emphasize different aspects, it is the interaction of these opposing forces and their comprehensive efforts that constitute the vitality, availability and lofty value of mathematical science. [1] object
The knowledge and application of basic mathematics is an indispensable part of individual and group life. The refinement of its basic concepts can be found in ancient mathematical documents of ancient Egypt, Mesopotamia and ancient India. Since then, its development has been very small.
Three-dimensional structure diagram
Until the Renaissance in16th century, the mathematical innovation generated by the interaction with new scientific discoveries led to the acceleration of knowledge, until today.
Today, mathematics is used in different fields of the world, including science, engineering, medicine and economics. The application of mathematics in these fields is usually called applied mathematics, and sometimes it will lead to new mathematical discoveries and the development of new disciplines. Mathematicians also study pure mathematics, that is, mathematics itself, without any practical application. Although many studies started from pure mathematics, many applications will be found later.
The French Bourbaki School, founded in 1930s, believes that mathematics, at least pure mathematics, is a theory to study abstract structures. Structure is a deductive system based on initial concepts and axioms. According to Boone School, there are three basic abstract structures: algebraic structure (group, ring, field, lattice …), ordered structure (partial order, total order …) and topological structure (neighborhood, limit, connectivity, dimension …).
field
The need of mathematical business calculation, understanding the system between numbers, measuring land area and predicting astronomical concepts. These four requirements are generally related to a wide range of mathematical fields (namely, arithmetic, algebra, geometry, analysis) such as quantity, structure, space and change. In addition to the above-mentioned main concerns, there are sub-fields used to explore the relationship between the core of mathematics and other fields: to logic, to set theory (foundation), to empirical mathematics in different sciences (applied mathematics), and to the rigorous study of uncertainty in modern times.
phrase
Mathematical mathematics; ; Mathematics; teacher
Mathematical analysis [number] mathematical analysis; Analysis; Mathematical analysis; Mathematical analyst
[span] mathematical programming [number] mathematical programming; Mathematical planning; MP； Mathematical subordinate programming
Edit some concepts in this paragraph.
circumference ratio
Pi π
The study of quantity begins with numbers, which are familiar natural numbers and integers, as well as rational numbers and irrational numbers described in arithmetic.
Another research field is its size, which leads to cardinality and another infinite concept: Alev number, which allows meaningful comparison between the sizes of infinite sets.
The first person to find the value of pi by scientific method was Archimedes, and he got the π value accurate to two decimal places. When Liu Hui, a mathematician, annotated Nine Chapters Arithmetic, he used the secant circle method to find the approximate value of π. Draw a conclusion. Zu Chongzhi, a mathematician and astronomer, worked hard to calculate the value of pi (∏) to seven decimal places for the first time in the history of mathematics in the world, that is, between 3. 14 15926 and 3. 14 15927.
π is an infinite acyclic decimal and an irrational number.
structure
Many mathematical objects, such as sets of numbers and functions, have internal structures. The structural properties of these objects are discussed in groups, rings, bodies and other abstract systems that are themselves objects. This is the field of abstract algebra. Here is a very important concept, that is, vector, which is extended to vector space and studied in linear algebra. The study of vector combines three basic fields of mathematics: quantity, structure and space. Vector analysis extends it to the fourth basic field, namely change.
space
The study of space comes from geometry, especially Euclidean geometry. Trigonometry combines space and numbers,
geometric figure
Contains the famous Pythagorean theorem. Now the research on space is extended to high-dimensional geometry, non-Euclidean geometry and topology. Numbers and spaces play an important role in analytic geometry, differential geometry and algebraic geometry. In differential geometry, there are concepts such as fiber bundle and calculation on manifold. Algebraic geometry has the description of geometric objects such as polynomial equation solution set, which combines the concepts of number and space; There is also the study of topological groups, which combines structure and space. Lie groups are used to study space, structure and change.
Foundation and logic
Rotating surface
In order to understand the basis of mathematics, mathematical logic and set theory are developed. German mathematician Georg Cantor (1845- 19 18) initiated the set theory, boldly marched into infinity, provided a solid foundation for all branches of mathematics, and its own content was quite rich, put forward the existence of real infinity, and made inestimable contributions to the future development of mathematics. Cantor's work has brought a revolution to the development of mathematics. Because his theory transcended intuition, it was opposed by some great mathematicians at that time. Poincare also compared set theory to an interesting "morbid situation", and Poincare retorted that Cantor was "neurotic" and "went into a hell beyond numbers". Cantor is still full of confidence in these criticisms and accusations. He said: "My theory is rock solid, and anyone who opposes it will shoot himself in the foot.
Set theory gradually penetrated into all branches of mathematics at the beginning of the 20th century, and became an indispensable tool in analytical theory, measurement theory, topology and mathematical science. Hilbert, the greatest mathematician in the world in the early 20th century, spread Cantor's thoughts in Germany, calling him "a mathematician's paradise" and "the most amazing product of mathematical thoughts". British philosopher Bertrand Russell praised Cantor's works as "the greatest works that can be boasted in this era".
Mathematical logic focuses on putting mathematics on a solid axiomatic framework and studying the results of this framework. It is the birthplace of Godel's second incomplete theorem, which is perhaps the most widely spread achievement in logic-there is always a true theorem that cannot be proved. Modern logic is divided into recursion theory, model theory and proof theory, which are closely related to theoretical computer science.
Strict symbol
trigonometric function
In modern symbols, simple expressions can describe complex concepts. This image is generated by a simple equation.
Most of the mathematical symbols we use today were invented after16th century. Before that, mathematics was written in words, which was a hard procedure that would limit the development of mathematics. Today's symbols make mathematics easier to be controlled by experts, but beginners are often afraid of it. It is extremely compressed: several symbols contain a lot of information. Like music notation, today's mathematical symbols have clear grammar and information codes, so it is difficult to write them in other ways.
Mathematical language is also difficult for beginners. How to make these words have more accurate meanings than everyday language? Novices are also troubled. Words such as openness and domain have special meanings in mathematics. Mathematical terms also include proper nouns such as embryo and integrability. But these special symbols and terms are used for a reason: mathematics needs accuracy more than everyday language. Mathematicians call this requirement for linguistic and logical accuracy "rigor".
Stiffness is a very important and basic part of mathematical proof. Mathematicians hope that their reasoning and axioms of the definite reason system can be inferred. This is to avoid the wrong "theorem", relying on unreliable intuition, there have been many examples in history. The rigor expected in mathematics changes with time: the Greeks expected careful argumentation, but in Newton's time, the methods used were not so rigorous. Newton's definition of solving problems was not carefully analyzed and formally proved until the19th century. Today, mathematicians have been arguing about the rigor of computer-aided proof. When a large number of measurements are difficult to verify, it is hard to say that they are effective and rigorous. Because of the differences of the times, a lot of knowledge has been erased, but mathematics will never be erased, and wisdom will always be circulated.
Edit this brief history of mathematics.
The origin of mathematics
Mathematics originated from early human production activities. As one of the six ancient arts in China (called "number" in the six arts), it was also regarded as the starting point of philosophy by ancient Greek scholars. Mathematics (mathematics; Greek: μ α θ η μ α κ? ) means "the basis of learning" and comes from μαρθξμα(máthema) ("science, knowledge and learning").
Evolution of mathematics
Addition comes from simple abstract concepts.
The evolution of mathematics can be regarded as the continuous development of abstraction and the extension of subject matter. The first abstract concept is probably number, and its cognition that two apples and two oranges have something in common is a great breakthrough in human thought. In addition to knowing how to calculate the number of actual substances, prehistoric people also knew how to calculate the number of abstract substances, such as time-date, season and year. Arithmetic (addition, subtraction, multiplication and division) will naturally occur. Ancient stone tablets also confirmed the knowledge at that time [2].
In addition, writing or other systems that can record numbers are needed, such as Mu Fu or chips used by the Inca Empire to store data. There are many different counting systems in history.
Since the historical era, the main principles in mathematics have been formed to do many calculations related to taxation and trade, understand the relationship between numbers, measure land and predict astronomical events. These needs can be simply summarized as the study of quantity, structure, space and time in mathematics.
elementary mathematics
(Lu. 1 math)
By16th century, elementary mathematics, such as arithmetic, elementary algebra and trigonometry, had been basically completed.
Advanced mathematics
(Lu. 2 mathematics)
The appearance of the concept of variables in the17th century made people begin to study the relationship between variables and the mutual transformation between graphs. In the process of studying classical mechanics, the method of calculus was discovered. With the further development of natural science and technology, set theory and mathematical logic, which are produced for studying the basis of mathematics, have also begun to develop slowly.
Edit this passage about the history of Chinese mathematics.
Mathematics, called arithmetic in ancient times, is an important subject in ancient Chinese science. According to the characteristics of the development of ancient mathematics in China, it can be divided into five periods: the germination period; The formation of the system; Development; Prosperity and the integration of Chinese and western mathematics.
seed
At the end of primitive commune, after the emergence of private ownership and commodity exchange, the concepts of number and shape developed further. The pottery unearthed during Yangshao culture period has been engraved with the symbol representing 1234. By the end of the primitive commune, written symbols had begun to replace knotted notes.
Pottery unearthed in Xi 'an Banpo has an equilateral triangle composed of 1 ~ 8 dots, and a pattern of 100 small squares divided into squares. The houses in Banpo site are all round and square. In order to draw circles and determine straightness, people have also created drawing and measuring tools such as rulers, moments, rulers and ropes. According to Records of Historical Records Xia Benji, Yu Xia used these tools in water conservancy.
In the middle of Shang Dynasty, a set of decimal numbers and notation had been produced in Oracle Bone Inscriptions, the largest of which was 30 thousand; At the same time, the Yin people recorded the date of 60 days with 60 names, including Jiazi, Yechou, Bingyin and Dingmao, which were composed of ten heavenly stems and twelve earthly branches. In the Zhou Dynasty, eight kinds of things were previously represented by eight diagrams composed of yin and yang symbols, which developed into sixty-four hexagrams, representing sixty-four kinds of things.
Zhouyi suan Jing
The book Parallel Computation in 1 century BC mentioned the methods of using moments of high, deep, wide and distance in the early Western Zhou Dynasty, and listed some examples, such as hook three, strand four, chord five and ring moments can be circles. As mentioned in the Book of Rites, the aristocratic children of the Western Zhou Dynasty had to learn numbers and counting methods from the age of nine, and they had to be trained in manners, music, shooting, bending, calligraphy and counting. As one of the "six arts", number has begun to become a special course.
During the Spring and Autumn Period and the Warring States Period, calculation has been widely used and decimal notation has been used, which is of epoch-making significance to the development of mathematics in the world. During this period, econometrics was widely used in production, and mathematics was improved accordingly.
The contention of a hundred schools of thought in the Warring States period also promoted the development of mathematics, especially the dispute of rectifying the name and some propositions were directly related to mathematics. Famous experts believe that the abstract concepts of nouns are different from their original entities. They put forward that "if the moment is not square, the rules cannot be round", and defined "freshman" (infinity) as "nothing beyond the maximum" and "junior" (infinitesimal) as "nothing within the minimum". He also put forward the idea that "one foot is worth half a day, which is inexhaustible".
Mohism believes that names come from things, and names can reflect things from different sides and depths. Mohist school gave some mathematical definitions. Such as circle, square, flat, straight, sub (tangent), end (point) and so on.
Mohism disagreed with the proposition of "one foot" and put forward the proposition of "non-half" to refute: if a line segment is divided into two halves indefinitely, there will be a non-half, which is a point.
The famous scholar's proposition discusses that a finite length can be divided into an infinite sequence, while the Mohist proposition points out the changes and results of this infinite division. The discussion on the definition and proposition of mathematics by famous scholars and Mohists is of great significance to the development of China's ancient mathematical theory.