First, the matching method
Matching method is a technique of directional deformation of mathematical formula (matching into a "complete square"), and the relationship between known and unknown is found through the formula, thus simplifying the complex. When formulating the formula, it is necessary to make appropriate predictions and rationally use the skills of "dividing", "adding", "matching" and "gathering" to complete the formula. Sometimes called "matching method".
The most common formula is identical deformation, so that the mathematical formula appears completely square. It is mainly suitable for the discussion and solution of known or unknown quadratic equations, quadratic inequalities, quadratic functions and quadratic algebra, or the translation transformation of quadratic curves without xy terms.
Second, alternative methods.
When solving a mathematical problem, we regard a formula as a whole and replace it with a variable, thus simplifying the problem. This is called substitution. The essence of substitution is transformation, the key is to construct elements and set elements, and the theoretical basis is equivalent substitution. The purpose is to change the research object, move the problem to the knowledge background of the new object, standardize non-standard problems, simplify complex problems and become easy to deal with.
Substitution method is also called auxiliary element method and variable substitution method. By introducing new variables, scattered conditions can be linked, implicit conditions can be revealed, or conditions can be linked with conclusions. Or turn it into a familiar form to simplify complicated calculation and derivation.
It can transform high order into low order, fraction into algebraic expression, irrational expression into rational expression, transcendental expression into algebraic expression, and has a wide range of applications in the study of equations, inequalities, functions, sequences, triangles and other issues.
Third, the undetermined coefficient method
The method of determining the functional relationship between variables, setting some unknown coefficients, and then determining these unknown coefficients according to given conditions is called undetermined coefficient method, and its theoretical basis is polynomial identity, that is, the necessary and sufficient conditions for using polynomial f(x)g(x) are: for any value of a, there is f (a) g (a); Or the coefficients of similar terms of two polynomials are correspondingly equal.
The key to solving the problem by undetermined coefficient method is to list the equality or equation correctly according to what is known. Using the undetermined coefficient method is to transform a mathematical problem with a certain form into a set of equations by introducing some undetermined coefficients. Judging whether a problem is solved by undetermined coefficient method mainly depends on whether the solved mathematical problem has a certain mathematical expression, and if so, it can be solved by undetermined coefficient method. For example, factorization, fractional decomposition, summation of series, finding functions, finding complex numbers, finding curve equations in analytic geometry, etc. These problems have clear mathematical expressions and can be solved by undetermined coefficient method.
Using the undetermined coefficient method, the basic steps to solve the problem are:
The first step is to determine the analytical formula of the undetermined coefficient problem;
Secondly, according to the condition of identity, a group of equations with undetermined coefficients are listed.
The third step is to solve the equations or eliminate the undetermined coefficients, so that the problem is solved.
How to list a set of equations with undetermined coefficients is mainly analyzed from the following aspects:
(1) Use the column equation with equal corresponding coefficients;
(2) Substituting the concept of identity into the normal equation by numerical value;
(3) Use the self-defined attribute sequence equation;
(4) Using geometric conditions to set equations.
For example, when solving the conic equation, we can use the undetermined coefficient method to solve the equation: first, set the form of the equation, which contains undetermined coefficients; Then the geometric conditions are transformed into equations or equations with unknown coefficients; Finally, the unknown coefficients are obtained by solving the obtained equations or equations, and the equations of conic curves are obtained by substituting the obtained coefficients into well-defined equation forms.
Fourth, the definition method
The so-called definition method is to solve problems directly with mathematical definitions. Theorems, formulas, properties and laws in mathematics are all derived from definitions and axioms. Definition is a logical method to reveal the connotation of a concept, which is clarified by pointing out the essential attributes of things reflected by the concept.
Definition is the inevitable result of thousands of practices, which scientifically reflects and reveals the essential characteristics of things in the objective world. Simply put, definition is a high abstraction of basic concepts by mathematical entities. Solving problems by definition is the most direct method. Let's go back to definition in this class.
Fifth, mathematical induction.
Induction is a way of thinking, which leads to general principles with special examples. Inductive reasoning can be divided into complete inductive reasoning and incomplete inductive reasoning. Incomplete inductive reasoning only infers the properties of a class of things according to the same properties of some objects, which is not allowed in mathematical reasoning. Complete inductive reasoning is to draw a conclusion after investigating all the objects of a class of things.
Mathematical induction is a reasoning method used to prove some mathematical propositions related to natural numbers, which is widely used to solve mathematical problems. It is a recursive mathematical proof method. The first step of argument is to prove that the proposition holds when n= 1 (or n), which is the basis of recursion. The second step is to assume that the proposition holds when n=k, and then prove that the proposition also holds when n=k+ 1 This is the theoretical basis of infinite recursion. It judges whether the correctness of a proposition can be generalized from special to general. In fact, it is to make the correctness of the proposition break through the limit and reach infinity. These two steps are closely related and indispensable. After completing these two steps, we can conclude that "the conclusion is correct for any natural number (or n≥n and n∈N)". It can be seen from these two steps that mathematical induction is a complete recursive induction.
When proving a problem by mathematical induction, the key is to deduce the proposition of n=k+ 1. The proof of this step should have a sense of purpose, and pay attention to the analysis and comparison with the final goal of solving problems, so as to determine and standardize the direction of solving problems, gradually narrow the differences, and finally achieve the purpose of solving problems.
The following problems can be proved by mathematical induction: identity, algebraic inequality, trigonometric inequality, sequence problem, geometry problem, divisibility problem and so on.
Six, parameter method
Parametric method refers to the process of solving problems by introducing some new variables (parameters) related to the mathematical objects studied in the subject as a medium, and then analyzing and synthesizing them. The parametric equation of straight line and conic is an example of solving problems by parametric method. The substitution method is also a typical example of introducing parameters.
Dialectical materialism affirms that the connection between things is infinite and the ways of connection are rich and varied. The task of science is to reveal the internal relations between things, so as to discover the changing laws of things. The function of parameters is to describe the changing state of things and reveal the internal relations between changing factors. Parameters reflect the idea of movement and change in modern mathematics, and their views have penetrated into all branches of middle school mathematics. It is common to solve problems by parameter method.
The key to solving problems by parameter method is to introduce parameters properly, communicate the internal relationship between known and unknown, and use the information provided by parameters to answer questions smoothly.
Seven. reductio ad absurdum
Different from the above methods, reduction to absurdity belongs to the category of "indirect proof", which is a proof method of thinking about problems from a negative perspective, that is, affirming the topic and denying the conclusion, thus leading to contradictory reasoning. Hadamard, a French mathematician, summed up the essence of reduction to absurdity: "If we affirm the hypothesis of the theorem and deny its conclusion, it will lead to contradictions". Specifically, reduction to absurdity is to start with the conclusion of negative proposition, take the negation of proposition conclusion as the known condition of reasoning, and make correct logical reasoning, so as to compare it with known conditions, known axioms, theorems, laws or propositions that have been proved to be correct. The reason for the contradiction is that the hypothesis is not established, so the conclusion of the proposition is affirmed and the proposition is proved.
Reduction to absurdity is based on the "law of contradiction" and "law of excluded middle" in the laws of logical thinking. In the same thinking process, two contradictory judgments cannot be true at the same time, and at least one of them is false, which is the "law of contradiction" in logical thinking; Two contradictory judgments cannot be false at the same time. Simply saying "one or not one" is the "law of excluded middle" in logical thinking. In the process of proving absurdity, contradictory judgments are obtained. According to the law of contradiction, these contradictory judgments cannot be true at the same time, but one of them must be false. But the known conditions, axioms, theorems, rules or propositions that have been proved to be correct are all true, so the "negative conclusion" must be false. According to "law of excluded middle", the contradictory and mutually negative judgments of conclusion and negative conclusion cannot be false at the same time, but there must be a truth, so we get that the original conclusion must be true. Therefore, reduction to absurdity is based on the basic laws and theories of logical thinking, and reduction to absurdity is credible.
The problem model of reduction to absurdity can be simply summarized as "negation → reasoning → negation". That is to say, starting from the negative conclusion, through correct reasoning, logical contradictions are led out and new negation is achieved. It can be considered that the basic idea of reduction to absurdity is "negation of negation". The three main steps of proof by reduction to absurdity are: denying the conclusion → deducing the contradiction → establishing the conclusion. The specific steps of implementation are:
The first step, reverse design: make assumptions contrary to the verification conclusion;
The second step is to return to absurdity: under the condition of reverse assumption, the contradiction is deduced through a series of correct reasoning;
The third step, conclusion: it shows that the reverse hypothesis is not established, thus affirming the original proposition.
When applying reduction to absurdity, we must use "counter-hypothesis" for reasoning, otherwise it is not reduction to absurdity. When proving a problem by reducing to absurdity, if only one aspect of the proposition needs to be proved, we only need to refute this situation. This reduction to absurdity is also called "reduction to absurdity". If the conclusion is multifaceted, then all the negative situations must be refuted one by one in order to infer the original conclusion. This method of proof is also called "exhaustive method".
Reduction to absurdity is often used to solve mathematical problems. Newton once said, "Reduction to absurdity is one of the most skilled weapons for mathematicians". Generally speaking, the problems commonly proved by reduction to absurdity are: the proposition that the conclusion appears in the form of "negative form", "at least" or "at most", "unique" and "infinite"; Or the negative conclusion is more obvious. Specific and simple proposition; Or directly prove the difficult proposition, change its thinking direction, and think negatively from the conclusion, and the problem may be solved very simply.