- Chinese name
- Mathematical method
- Features
- High abstraction and generality
- Definition
- Use mathematical language to express the state, relationship and process of things, and deduce, calculate and analyze them, so as to form the method of problem interpretation, judgment and prediction
Mathematical methods have the following three basic characteristics:
First, high abstraction and General ； Second Accuracy That is, logical rigor and conclusive certainty ； Third, applied universality And operability. Mathematical methods play an important role in scientific and technological research: first, provide concise and accurate formal language, second, provide quantitative analysis and calculation methods, and third, provide logical reasoning Tools. Modern science and technology In particular, the development of electronic computers and the strengthening of the position and role of mathematical methods complement each other. stay Middle School Mathematics The basic mathematical methods often used in can be roughly divided into the following three categories: for example analytical method (including counter evidence method) synthesis 、 Counterevidence method 、 Induction 、 Exhaustion method (require classified discussion), etc. These methods should comply with logic The basic laws and rules in the "" have the characteristics of mathematics because they are applied to mathematics. Such as modeling Elimination method , reduction method Substitution method 、 Image method (Also called Coordinate method , often called image method in algebra analytic geometry It is often called coordinate method), comparison method (mainly referring to comparison of size in mathematics, which is different from multi-directional comparison in logic) Scaling method , and vector method to be learned in the future Mathematical induction (This is different from Incomplete induction These methods are extremely important and widely used. for example Matching method 、 Undetermined coefficient method , elimination method Formula method 、 Substitution method (also known as intermediate variable method), item splitting and item supplementing method (including adding auxiliary elements to realize reduction) Mathematical thought )、 Factorization Methods, and Parallel moving method , folding method, etc. These methods also play an important role in solving some mathematical problems. Whether natural science Technical science or social sciences In order to gain a deeper understanding of the quality of the objects studied, we need to describe them in terms of quantity, which requires mathematical methods. For things of different nature and complexity, the requirements and possibilities of applying mathematical methods are different. On the whole, a science is really mature only when it can use mathematics. stay modern science The degree of using mathematics has become an important symbol to measure the development of a science, especially the maturity of its theory. The key to the successful application of mathematical methods in scientific research is to extract a suitable mathematical model This model can not only reflect the essence of the problem, but also make the problem get the necessary simplification to facilitate mathematical derivation. Establishing mathematical model is to carry out make a concrete analysis Of Scientific abstraction Therefore, we should be good at grasping the main contradictions, highlighting the main factors and relationships, and setting aside those Secondary factors And relationships. The process of modeling is also a process of "simplifying the complexity" and "simplifying the difficulty". Of course, simplification is not unconditional. Reasonable simplification must take into account what practical problems allow Error range And the preconditions required by the mathematical method used. For the same problem, different mathematical models can be established. At the same time, the best model can be gradually selected through continuous testing and comparison in the research process, and it can be continuously tested and modified in the application process to gradually improve it. The mathematical model abstracted from a special problem often has a certain degree of universality, because a special mathematical model can develop into a common mathematical model describing the same kind of phenomenon. There are generally two types of mathematical models that have been widely used and fruitful: one is called Deterministic model , that is, using various mathematical equations such as algebraic equation differential equation 、 integral equation 、 Difference equation To describe and study various Inevitability Phenomenon, in this kind of model, the change and development of things follow certain mechanics Regularity ； The other is called Randomness model , ready to use probability theory And mathematical physics statistical method Describe and study various probability phenomena, and the development and change of things are shown as Randomness Process and follow Statistical law , and has Multiple possibilities Results. objective world The inevitability phenomenon and the probability phenomenon are not completely separated. Some things are mainly inevitable, but when the influence of random factors cannot be ignored, it is necessary to Deterministic model The introduction of random factors into the stochastic differential equation Such a mathematical model. Since the 1970s, it has been found in some deterministic models, such as some descriptions Conservative system or dissipation Structural nonlinear equation There is no additional random factor, but it shows "internal randomness" within a certain parameter range, that is, there are bifurcation and chaotic random behaviors. The mechanism of this kind of phenomenon and its mathematical problems have attracted the attention of mathematicians and scientists and are being studied. Mathematics itself is constantly developing, and the research on the relationship between various quantities, quantities and changes in quantities is also increasingly deepening Mathematical concept New mathematical branches are constantly emerging, and new mathematical methods are also breeding and sprouting correspondingly. With the increasingly extensive penetration of mathematics into various sciences and its combination with various objects and problems, people are refining various new mathematical models from them and creating various new Mathematical tools 。 Especially electronic computer The application of the mathematical method showed new vitality, and the so-called "mathematical experimental method" appeared. The essence of this method is not to experiment on the actual object, but to "experiment" on its mathematical model. The operation of this "experiment" is to realize a large number of numerical operations and Logical operation 。 This makes it possible to complete the trial calculation subject that was difficult to carry out because of the heavy workload in the past. The development prospect of mathematical methods in this field is considerable. 
