Mathematics is a science that studies quantitative relations and spatial forms in the real world. Simply put, it is the science of studying numbers and shapes. Due to the needs of life and labor, even the most primitive people know simple counting, and it has developed from counting with fingers or objects to counting with numbers. In China, the method of expressing large numbers with decimals appeared at the latest in Shang Dynasty. By the Qin and Han Dynasties, there was a perfect decimal system. In "Nine Chapters Arithmetic", not later than the first century A.D., it contains the calculation rules of square roots and square roots that are only possible in numerical systems, as well as various operations of fractions and methods for solving linear simultaneous equations, and also introduces the concept of negative numbers.
Liu Hui also proposed to use decimals to represent the odd zero part of the square root of irrational numbers in Annotations to Nine Arithmetic Chapters, but it was not until the Tang and Song Dynasties (in Europe, after Steven in16th century) that decimals were widely used. In this book, Liu Hui approximated the circumference of a circle with the circumference of a regular polygon inscribed in the circle, which became a common method for later generations to find pi.
Although China never had a general concept of irrational numbers or real numbers, in essence, China had already completed all the arithmetic and methods of the real number system at that time, which was indispensable not only in application, but also in early mathematics education. As for Europe, which inherited the culture of Babylon, Egypt and Greece, it focuses on the study of the nature of numbers and the logical relationship between these properties.
As early as Euclid's Elements of Geometry, there were some conclusions such as the concept of prime number, the infinite number of prime numbers and the unique decomposition of integers. Numbers without fractions were discovered in ancient Greece and are now called irrational numbers. Since16th century, complex numbers have appeared again because of solving higher-order equations. In modern times, the concept of number was further abstracted. According to the different operation rules of number, the general number system was discussed independently in theory, forming several different branches of mathematics.
Square root and square root are necessary operations to solve the simplest higher order equation. In "Nine Chapters Arithmetic", a special form of quadratic equation was solved. In the Song and Yuan Dynasties, a clear concept of "Tianyuan" (i.e. unknown number) was introduced, and the methods of finding numerical solutions of higher-order equations and simultaneous solutions of higher-order algebraic equations with up to four unknowns appeared, commonly known as "Tianyuan" and "Quaternary". The expression, algorithm and elimination method of polynomial are close to modern algebra.
Outside China, the works of Hua ramiz, an Arab in the 9th century, elaborated the solution of quadratic equation, which is generally regarded as the originator of algebra, and its solution is essentially the same as the geometric method relying on cutting in ancient China. Ancient mathematics in China devoted itself to the concrete solution of equations, while European mathematics originated from ancient Greece and Egypt was different, and generally devoted itself to exploring the properties of equation solutions.
/kloc-In the 6th century, the Vedas used words instead of equation coefficients and introduced algebraic symbolic calculus. Exploring the properties of algebraic equations is the emergence of concepts and theories such as determinant, matrix, linear space and linear transformation derived from linear equations. From algebraic equations leading to the introduction of complex numbers and symmetric functions, to the establishment of Galois theory and group theory. Algebraic geometry, which is extremely active in modern times, is nothing more than a theoretical study of the set of solutions of higher-order algebraic equations.
The study of shape belongs to the category of geometry. Ancient peoples all had simple concept of shape, which was often expressed by pictures, and the reason why graphics became mathematical objects was due to the requirements of tool making and measurement. Rules are used as squares. In ancient China, Yu Xia had measuring tools such as ruler, moment, ruler and rope when berthing on the water.
Mo Jing has a series of abstract generalizations and scientific definitions of geometric concepts. Zhou Kuai's Su 'an Classic and Liu Hui's Island Su 'an Classic give general methods and specific formulas for observing heaven and earth with moments. In Liu Hui's Nine Chapters Arithmetic and Nine Chapters Arithmetic, besides Pythagorean Theorem, some general principles for solving various problems are put forward. For example, the principle of finding the area of any polygon is complementary; The two-to-one principle (Liu Hui principle) needed to find the volume of polyhedron; In the 5th century, in order to find the volume of a curve, especially the volume of a sphere, Zu (Riheng) put forward the principle that "if the potentials are the same, the products cannot be different". There is also a limit method (secant), which uses inscribed regular polygons to approximate the circumference of a circle. However, since the Five Dynasties (about 10 century), China has made little achievements in geometry.
China's geometry takes the calculation of area and volume as the central task, while the ancient Greek tradition attaches importance to the relationship between the nature of shape and various properties. Euclid's Elements of Geometry established a deductive system consisting of definitions, axioms, theorems and proofs, which became a model of axiomatization of modern mathematics and influenced the development of the whole mathematics. Especially, the study of parallel axioms led to the appearance of non-Euclidean geometry in19th century.
Since the Renaissance in Europe, through the study of the perspective relationship of painting, projective geometry has emerged. In the18th century, gaspard monge applied analytical methods to study shapes and created differential geometry. Gauss's surface theory and Riemann's manifold theory create a research method that regards shape as an independent object without surrounding space. /kloc-In the 9th century, Klein unified geometry from the point of view of groups. In addition, such as Cantor's point set theory, it expands the scope of shapes; Poincare founded topology, making the continuity of shape the object of geometric research. All these make geometry look brand-new.
In the real world, numbers and shapes are inseparable like shadows. China's ancient mathematics reflected this objective reality, and numbers and shapes always complemented each other and developed in parallel. For example, Pythagorean survey puts forward the requirement of square root, and the method of square root and square root is based on geometric considerations. The generation of quadratic and cubic equations also mostly comes from geometric and practical problems. In Song and Yuan Dynasties, due to the introduction of Tianyuan concept and equivalent polynomial concept, geometric algebra appeared.
In astronomy and geography, the drawing of catalogues and maps has used numbers to represent places, but it has not developed to the point of coordinate geometry. In Europe, by the14th century, Ao' Erslmu's works on the graphic representation and function of latitude and longitude had sprouted. 17th century, Descartes put forward a systematic method of algebraic representation of geometric things and its application. Inspired by it, through the work of Leibniz and Newton, it developed into a modern form of coordinate analytic geometry, which made the unification of numbers and shapes more perfect. It not only changed the old geometric proof method that followed Euclid geometry in the past, but also caused the generation of derivatives, which became the root of calculus. This is a great event in the history of mathematics.
17th century, due to the requirements of science and technology, mathematicians studied movement and change, including the change of quantity and the transformation of shape (such as projection), and also produced the concepts of function and infinitesimal analysis, which is now calculus, which made mathematics enter a new era of studying variables.
/kloc-Since the 8th century, with the creation of analytic geometry and calculus as an opportunity, mathematics has developed rapidly on an unprecedented scale, and many branches have emerged. Because the objective laws of nature are mostly expressed in the form of differential equations, the study of differential equations has received great attention from the beginning.
Differential geometry and calculus were born at the same time, and the work of Gauss and Riemann produced modern differential geometry. 19 at the turn of the 20th century, poincare founded topology, which opened up a way to study continuous phenomena qualitatively and integrally. The analysis of random phenomena in the objective world produces probability theory. The military needs of the Second World War, as well as the complexity of large-scale industry and management, gave birth to operations research, system theory, cybernetics, mathematical statistics and other disciplines. Practical problems need specific numerical solutions, which leads to computational mathematics. The requirement of choosing the best way has produced various optimization theories and methods.
The development of mechanics, physics and mathematics has always influenced and promoted each other, especially relativity and quantum mechanics, which promoted the growth of differential geometry and functional analysis. In addition, in the19th century, only one equation was used in chemistry, and mathematics was hardly used in biology, and some cutting-edge mathematical knowledge was already used.
/kloc-In the late 9th century, set theory appeared and entered a critical era, which promoted the formation and development of mathematical logic, and also produced various ideological trends and basic schools of mathematics that regarded mathematics as a whole. Especially in 1900, the German mathematician Hilbert gave a speech on the important issues of contemporary mathematics at the second international congress of mathematicians. The rise of the French Bourbaki school, which was initiated in the 1930s to treat mathematics with the concept of structure, had a great and far-reaching impact on the development of mathematics in the twentieth century, and the word scientific mathematicization began to be enjoyed by people.
The edge of mathematics continues to penetrate and expand into natural science, engineering technology and even social science, and draws nutrition from it, and some marginal mathematics appears. The inherent demand of mathematics itself has also spawned many new theories and branches. At the same time, its core part is constantly consolidated and improved, and sometimes it is properly adjusted to meet external needs. In a word, the big tree of mathematics is flourishing and has deep roots.
In the vigorous development of mathematics, the concepts of number and shape are constantly expanding and becoming more and more abstract, so that there are no traces of initial counting and simple graphics. Nevertheless, in the new branch of mathematics, there are still some objects and operational relationships expressed by geometric terms. For example, treat a function as a point in a certain space. In the final analysis, this method is effective because mathematicians have been familiar with simple mathematical operations and graphic relations, and have a long-term and profound practical foundation. Moreover, even the most primitive numbers, such as 1, 2, 3, 4, and geometric figures, such as points and straight lines, have been highly abstracted. Therefore, if number and shape are understood as generalized abstract concepts, then the above-mentioned definition of mathematics as a scientific study of number and shape is also applicable to modern mathematics at this stage.
Because the quantitative relationship and spatial form of mathematical research objects come from the real world, although mathematics is highly abstract in form, it is always rooted in the real world. Life practice and technical demand are always the real source of mathematics. Conversely, mathematics plays an important and key role in the practice of transforming the world. The enrichment, perfection and wide application of theory have always been accompanied and promoted each other in the history of mathematics.
However, due to the different objective conditions of different nationalities and regions, the specific development process of mathematics is also different. Generally speaking, the ancient Chinese people used bamboo as a calculation tool, which naturally produced a decimal numerical system. The superiority of the calculation method is helpful to the concrete solution of practical problems. The mathematics developed from this has formed a unique system characterized by constructiveness, computability, programmability and mechanization. The main goal is to start from the problem and then solve it. Ancient Greece paid attention to thinking and pursued the understanding of the universe. From this, it has developed into an axiomatic deductive system with abstract mathematical concepts and properties and their logical interdependence as the research object.
After China's mathematical system reached its peak in Song and Yuan Dynasties, it began to stagnate and almost disappeared. In Europe, a series of changes, such as the Renaissance, the religious revolution and the bourgeois revolution, led to the industrial revolution and the technological revolution. The use of machines has a long history at home and abroad. However, in China, it was banned by the emperors in the early Ming Dynasty, who thought it was a strange skill.
In Europe, it is developed because of the development of industry and commerce and the stimulation of navigation. Machines liberate people from heavy manual labor and lead them to theoretical mechanics and general scientific research on movement and change. Mathematicians at that time actively participated in these changes and the solution of corresponding mathematical problems, and produced positive results. The birth of analytic geometry and calculus has become a turning point in the development of mathematics. /kloc-the mathematical leap since the 0/7th century can generally be regarded as the continuation and development of these achievements.
In the 20th century, all kinds of brand-new technologies appeared, resulting in a new technological revolution, especially the appearance of electronic computers, which made mathematics face a new era. A feature of this era is the gradual mechanization of some mental work. Unlike17th century, mathematics has been dominated by concepts such as continuity and limit. Due to the needs of computer development and application, discrete mathematics and combinatorial mathematics have been paid attention to.
The role of computer in mathematics is not limited to numerical calculation, but also involves symbolic operation (including mathematical research such as machine proof). In order to better cooperate with computers, the requirements of mathematics for constructiveness, computability, programmability and mechanization are also quite prominent.
For example, algebraic geometry is a highly abstract mathematics, and the recent formulation of calculating algebraic geometry and constructing algebraic geometry is one of its clues. In a word, mathematics develops with the revolution of new technology.