Rosa Peter, a famous Hungarian mathematician, explained how mathematicians solved problems by conversion through a very vivid and interesting joke in his masterpiece Infinite Things. Someone asked, "Suppose you have a gas stove, faucet, kettle and matches in front of you. What do you want to do if you want to boil water?" Someone replied, "Fill the kettle with water, light the gas, and then put the kettle on the gas stove." The questioner affirmed the answer, but he asked, "What should you do if other conditions have not changed, but there is enough water in the kettle?" At this time, the questioner will definitely answer loudly and confidently, "Light the gas and put the kettle on." But a more perfect answer should be like this: "Only physicists will do what they just said, and mathematicians will answer," As long as the water in the kettle is emptied, the problem boils down to the above problems. "。
"Pour out the water", that is, the conversion method, which is commonly used by mathematicians. Turning to the history of mathematics development, there are countless examples, and the famous problem of the Seven Bridges in Konigsberg is a wonderful example. Great mathematician Euler's thinking program to solve this problem is:
This is a good application of transformation problem, from which we can easily sum up the thinking mode of transformation thinking method:
It can be seen that the strength of problem-solving ability lies in: 1, only keen insight can find the target model, and 2, only strong reduction ability can effectively transform the problem into the target model, and it is easier to solve it by using the inherent laws of the model.
In middle school mathematics, the common basic reduction forms are:
1, conversion between numbers. For example, calculate a formula to get a numerical value; Simplify an analytical formula to get the result; Solve the equation given by deformation; Transform the given inequality, solve the set and the mutual transformation among functions, equations and inequalities.
2. Conversion between forms. For example, using the knowledge of image transformation to make functional images; Use division, shape filling, folding and expansion as auxiliary lines and surfaces to deal with space graphics or plane graphics. , including turning a three-dimensional problem into a plane problem.
Example 2. As shown in the figure, in the regular triangular pyramid P-ABC, the length of each side is 2, e is the midpoint of the side PC, and d is any point of the side PB. Find the minimum perimeter of △ADE.
3. Digital-to-shape conversion. Digital-to-shape conversion is mainly based on the relationship between function and its image; Complex numbers and the geometric meaning of their operations; And the concepts of curves and equations in analytic geometry.
[Analysis]: This is an inequality proof problem with four unreasonable formulas, which is difficult to start with, and the reduction method can be applied. I noticed that the structure of the four irrational number formulas on the left is similar to Pythagorean theorem, so I thought of it and tried to reduce it to a geometric problem. This can be easily simplified to one: construct a square as shown in Figure 3, and it can be said that the inequality relationship is self-evident.