What is the basic idea of mathematics?

The so-called mathematical thinking refers to the spatial form and quantitative relationship of the real world reflected in human consciousness, as well as the result of thinking activities. Mathematical thought is the essential understanding after summarizing mathematical facts and theories; The thought of basic mathematics is the basic, summative and most extensive mathematical thought embodied or should be embodied in basic mathematics. They contain the essence of traditional mathematical thought and the basic characteristics of modern mathematical thought, and are historically developed. Through the cultivation of mathematical thinking, the ability of mathematics will be greatly improved. Mastering mathematical thought means mastering the essence of mathematics. Function and Equation Thought Function thought refers to analyzing, transforming and solving problems with the concept and properties of function. The idea of equation is to start with the quantitative relationship of the problem, transform the conditions in the problem into mathematical models (equations, inequalities or mixed groups of equations and inequalities) with mathematical language, and then solve the problem by solving equations (groups) or inequalities (groups). Sometimes, functions and equations are mutually transformed and interrelated, thus solving problems. Descartes' equation thought is: practical problem → mathematical problem → algebraic problem → equation problem. The universe is full of equality and inequality. We know that where there are equations, there are equations; Where there is a formula, there is an equation; The evaluation problem is realized by solving equations ... and so on; The inequality problem is also closely related to the fact that the equation is a close relative. Column equation, solving equation and studying the characteristics of equation are all important considerations when applying the idea of equation. Function describes the relationship between quantities in nature, and the function idea establishes the mathematical model of function relationship by putting forward the mathematical characteristics of the problem, so as to carry out research. It embodies the dialectical materialism view of "connection and change". Generally speaking, the idea of function is to use the properties of function to construct functions to solve problems, such as monotonicity, parity, periodicity, maximum and minimum, image transformation and so on. We are required to master the specific characteristics of linear function, quadratic function, power function, exponential function, logarithmic function and trigonometric function. In solving problems, it is the key to use the function thought to be good at excavating the implicit conditions in the problem and constructing the properties of resolution function and ingenious function. Only by in-depth, full and comprehensive observation, analysis and judgment of a given problem can we have a trade-off relationship and build a functional prototype. In addition, equation problems, inequality problems and some algebraic problems can also be transformed into functional problems related to them, that is, solving non-functional problems with functional ideas. Function knowledge involves many knowledge points and a wide range, and has certain requirements in concept, application and understanding, so it is the focus of college entrance examination. The common types of questions we use function thought are: when encountering variables, construct function relations to solve problems; Analyze inequality, equation, minimum value, maximum value and other issues from the perspective of function; In multivariable mathematical problems, select appropriate main variables and reveal their functional relationships; Practical application of problems, translation into mathematical language, establishment of mathematical models and functional relationships, and application of knowledge such as functional properties or inequalities to solve them; Arithmetic, geometric series, general term formula and sum formula of the first n terms can all be regarded as functions of n, and the problem of sequence can also be solved by function method. The idea of the combination of numbers and shapes is that "numbers are invisible, not intuitive, and there are countless shapes, so it is difficult to be nuanced". The combination of numbers and shapes can make the problem to be studied difficult and simple. Combining algebra with geometry, such as solving geometric problems by algebraic method and solving algebraic problems by geometric method, is the most commonly used method in analytic geometry. For example, finding the root number ((A- 1)2+(B- 1)2)+ root number (A 2+(B- 1)2)+ root number ((a-1) 2+B. The idea of classification discussion should be one. Such as solving inequality | a-1| >; 4. It is necessary to discuss the value idea of a. When a problem may be related to an equation, we can construct an equation and study its properties to solve this problem. For example, when proving Cauchy inequality, Cauchy inequality can be transformed into a discriminant of quadratic equation. Holism is based on the overall nature of the problem, highlighting the analysis and transformation of the overall structure of the problem, finding the overall structural characteristics of the problem, being good at treating some formulas or figures as a whole with a "holistic" vision, grasping the relationship between them, and carrying out purposeful and conscious overall treatment. The holistic thinking method is widely used in simplification and evaluation of algebraic expressions, solving equations (groups), geometric proof and so on. Integral substitution, superposition multiplication, integral operation, integral demonstration, integral processing and geometric complement are all concrete applications of integral thinking method in solving mathematical problems. The idea of transformation is to transform unknown, unfamiliar and complex problems into known, familiar and simple problems through deduction and induction. Mathematical theories such as trigonometric function, geometric transformation, factorization, analytic geometry, calculus, and even rulers and rulers of ancient mathematics are permeated with the idea of transformation. Common transformation methods include: general special transformation, equivalent transformation, complex and simple transformation, number-shape transformation, structural transformation, association transformation, analogy transformation and so on. The idea of implicit conditions is not clearly expressed, but it can be inferred from the existing explicit expressions, or it is not clearly expressed, but it is a routine or a truth. Analogy is to compare two (or two) different mathematical objects. If they are found to have similarities or similarities in some aspects, it is inferred that they may have similarities or similarities in other aspects. In order to describe an actual phenomenon more scientifically, logically, objectively and repeatedly, people use a language that is generally considered rigorous to describe various phenomena. This language is mathematics. What is described in mathematical language is called a mathematical model. Sometimes we need to do some experiments, but these experiments often use abstract mathematical models as substitutes for actual objects, and the experiments themselves are also theoretical substitutes for actual operations. The idea of transformation is to turn the unknown into the known, simplify the complex and turn the difficult into the easy. For example, fractional equations are transformed into integral equations, algebraic problems are transformed into geometric problems, quadrilateral problems are transformed into triangular problems and so on. The methods to realize this transformation are: undetermined coefficient method, collocation method, whole replacement method, static transformation, from abstract to concrete and so on. The inference that all objects of such things have these characteristics, or the inference that general conclusions are summarized from individual facts, is called inductive reasoning (induction for short). In short, inductive reasoning is reasoning from part to whole, from individual to general. In addition, there are mathematical ideas such as probability and statistics. For example, probability statistics refers to solving some practical problems through probability statistics, such as winning the lottery, comprehensive analysis of an exam and so on. In addition, some area problems can be solved by probability method.

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