Let's discuss how to use mathematics to solve problems.
There are three ways to prove a proposition:
1, the conventional proof method, deduces a proposition from an axiom or a known proposition, that is, proves that the proposition is a sub-proposition of a known axiom. The key point is to clarify the meaning and conditions of the proposition and find out the equivalent meaning and conditions of the proposition. It is best to convert it into a numerical equality relationship, and then carry out symbolic calculation. This calculus method is universal, and in some special cases, it can also be transformed into intuitive geometric relations, which can be proved by intuitive geometric relations, but geometric methods need inspiration and are not universal.
2, reduction to absurdity, assuming that the proposition is not established, deducing the contradictory proposition, thus proving that the proposition is established. The application is limited, so I won't introduce it.
3. Recursion, the initial proposition holds. If the nth proposition holds, then the nth proposition also holds, thus proving that all propositions hold. This kind of proof has strong limitations, so I won't introduce it.
We use the most typical Pythagorean law to illustrate the conventional proof method of derivation: to prove the Pythagorean law,
Analysis process:
1. Explicit proposition to be proved: Pythagorean law is that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.
2. Clear definition: The definition of a right triangle is that one of the angles is a right angle.
3. Find the equivalent meaning and convert it into symbolic calculus:
4. The square formed by the sides is equivalent to the area of the square, so we can consider using the characteristics of the right triangle to splice the graphics. There are many splicing methods, but they are all ideas that can't be copied. I won't introduce it here.
5. To put it another way, since Pythagoras' law is the relationship between side length values, we can consider the unique characteristics of right triangle to make the relationship between side length values happen, express it with equations, and then calculate it mathematically and convert it into a square relationship. This way of thinking is suitable for any occasion, and everyone can master it step by step. The relationship between side length values can only be realized by the equal side length ratio of similar triangles. Therefore, similar triangles are considered, because the characteristics of right triangles can only be reflected by using right angles. It is natural to think that the auxiliary line perpendicular to the hypotenuse will be drawn from a right angle.
It is easy to prove that the two newly generated right-angled triangles are similar to the original large right-angled triangles, which is also the characteristic of right-angled triangles. To describe similarity with numerical equations, there are three more variables, c 1, c2 and H need three equations to be eliminated, and to derive the relationship between A, B and C, it needs a fourth equation, so total * * needs four equations:
The small triangle below is similar to the big triangle:
b/c = c2/b
h/a = b/c
The small triangle above is similar to the big triangle:
a/c = c 1/a
h/b = a/c
Take c 1, c2 and h as variables, use three of them arbitrarily, solve their expressions, and bring them into the fourth equation that has not been used yet. The conversion equation is:
A square +b square = c square
This method of relation equation calculus, also called equation method, is suitable for most occasions and is also the most important mathematical content. The application of equation method is not only to prove the proposition, but also to deduce the specific value of unknown quantity. In a simple case, only arithmetic thinking can solve it, but in a slightly complicated case, the equation is the only solution.
To use a formula:
1, find out the conditions in the topic, and give the meaning and implied value of the numerical value. Replace the unknown quantity of the required solution with easy-to-understand symbols, including the unknown quantity of the required solution and the unknown quantity that may be needed.
2. For a physical quantity, find the equation relationship between values in pairs until the number of equations is not less than the number of unknown quantities.
3. Transform equation with mathematical calculus rate, and add, subtract, multiply, divide, find root, differential integral, term expansion, etc. Until the equation relationship between a single unknown quantity and a specific value is solved.
Give an example of how to use this equation:
Example 1 (primary school math problem):
A pipeline project is constructed by two engineering teams of Party A and Party B, and independent construction takes 10 and 15 days. If two teams work for two days at the same time, how many days will it take Team B to complete the rest of the project alone?
First, use the direct arithmetic deduction method: the engineering quantity is 1, and the daily engineering quantity that Party A and Party B can complete is110,115. After two days of simultaneous construction, there is still 1-(2/ 10+2/655.
This derivation method requires a slightly complicated thinking process. Simple, can think from many angles. Complex, often only one idea is feasible, and it is impossible without thinking.
Now we use the method of equation, without thinking at all, and only consider the quantitative relationship, and then we can get the required answer through mathematical calculation. The quantitative relationship can be considered from different angles and is equivalent:
The number of days used is unknown, and the symbol is marked as X days.
Method 1: the workload completed in two days plus the workload completed by team b in x days is equal to 1, i.e.
2/ 10+2/ 15+x * 1/ 15 = 1
Method 2: The sum of the quantities completed by Party A and Party B is equal to 1, that is
2/ 10+(2+x)* 1/ 15 = 1
Method 3: The remaining workload is the workload completed by Group B within X days, namely
1-(2/ 10+2/ 15)= x * 1/ 15
It can be seen that the quantitative relationship can be described from different angles by equation method, which is very easy to think of, and then the solution can be obtained by regular calculus. But it is a little difficult to deduce directly with thinking, that is, arithmetic method. This example is a very simple application problem and can also be worked out by arithmetic. However, no matter how clever the brain is, no matter how many application problems there are, you can't think of arithmetic ideas. All quantitative relations can only be listed as equations by equation method, and then the column relations can not be omitted, otherwise the answer can not be made. Repeated relations will be found in the calculation, and the redundant relations can be directly removed without affecting the calculation.
Example 2, slightly difficult (or elementary school math problem):
A railway bridge is 1000 meters long, and the train passes on it. It takes 1 minute for the train to pass completely, and the whole train stays on the bridge for 40 seconds. Please ask the length and speed of the train.
It is difficult to come up with an arithmetic concept.
Current equation method: suppose the train speed is x m/s and the length is y m.
There are three values: bridge length 1000m, bridge crossing time 1min, and the whole train stays on the bridge for 40 seconds. We just need to consider the relationship in pairs.
Before 1000m and 1 min:1000 = 60 * x–y.
1000 meter 40 seconds or 1 minute 40 seconds, which pair is easy to express?
1000 = 40 * x+y or (60–40) * x = 2 * y.
Two of the three equations can completely describe the relationship, and all three are repeated (any two can derive the third relationship). If you can't judge whether it is duplicate, list it all. Anyway, it can be found in the operation, which will not affect the solution.
In view of these simple application problems, in fact, when we calculate equations or equations, each step of calculation has practical significance, but in the calculation of complex equations, most of each step of calculation has no practical physical significance and is purely the application of mathematical rules. Therefore, some profound physical problems may only be discovered and explained by mathematical methods.
What is emphasized here is the problem of transforming application problems into equations or equations, which is the key to solving problems. Set the value to be solved to symbols x, y, z, etc. Write the numerical value or implied numerical value and meaning in the topic, indicate the meaning, and then take out two numerical values to consider their relationship. For a physical quantity, introduce other quantities, list the quantitative relationship, that is, equations, until all the values are used, and then put several equations together and solve them with mathematical calculus. It doesn't matter if the equation is repeated a lot, it will be removed when calculus is found. This problem-solving step does not need to be clever, nor does it need to consider many situations at the same time. It only needs to consider the problems one by one, then list the relationships, and finally abandon the actual scene and only do mathematical operations.
Example 3, (high school knowledge level):
20 kilometers in front of the enemy position, the exit speed of our artillery 1000 m/s, what is the elevation angle of the gun barrel when attacking the enemy?
We can't figure out the answer with arithmetic thinking, so we can only use the method of equation.
The elevation angle is set to y, and there are two values here, 20km, 1000m/s, indicating its physical meaning. Then, find out the quantitative relationship in pairs and combine them at will. According to the physical meaning, the quantitative relationship must be the relationship between the same physical quantity.
Relationship between elevation Y and 20km distance: Considering the relationship of spatial distance, elevation X leads to the horizontal flight of 20km when the shell lands, so it is necessary to introduce flight time t, so the relationship is as follows:
1000 * cos(y) * t = 20，000
Relationship between 20km distance and 1000m/s speed: The distance relationship has been considered above, so only other physical quantities can be considered this time. In this example, the physical quantities involved include time and speed, which we can choose at will. If it is found to be equivalent to the listed relationship, we will change it to another one. The speed here is equivalent to the distance relationship mentioned above, so we can only choose time: the time of horizontal flight of 20km and the shell.
20000/(1000 * cos y) = 2 *1000 * sin y/g, where g is the gravitational acceleration of 9.8 m/s/s.
Two equations, two variables, according to the rules of mathematical calculus, it is easy to find the specific value of elevation y.
Example 4, (Senior High School Knowledge)
When the enemy shells attacked, our radar measured the positions of three flying shells at intervals of 1 sec: (x 1, y 1, z 1) = (20km, 10km, 10km), (. Z3) = (18km, 9.7km, 10km), and x, y and z represent horizontal position, height and lateral direction respectively. Ask the enemy where the artillery is.
First, make clear the meaning of position: the shell shoots at a certain elevation angle, flies under the action of gravity, and is captured by our radar at a certain moment, and three position coordinates are measured at the distance 1 s. Substitute symbols for unknowns. Suppose the position of enemy artillery is (X0 Y0, Z0), the required elevation angle is a, the speed of shell leaving the chamber is v, the time to fly to position 1 is t, the falling speed of shell at position 1 is V 1, and the falling speed at position 2 is V2.
Look at the lateral position relationship first:
X 1-X2=V * COS(a) * 1
X 1-X3=V * COS(a) * 2
X0-X 1=V*COS(a) * t
Look at the vertical position relationship:
Y1-y2 = 0.5 * v2 2/g-0.5 * v12/g.
Y 1-Y0=0.5*V 1^2/g
The relationship between the descending speed:
V2-V 1=g * 1
V 1= (t-V*SIN(a)/g)* g
There are seven unknowns and seven relational equations, so we can get seven unknowns. If the z values of the three positions are different, we will list some equations of lateral position relationship in the z direction, and the elevation angle should be decomposed into the included angle between two planes. The equation is only a little more complicated, and the value of Z0 can also be obtained. In this way, the position (X0, Y0, Z0) of the enemy artillery can be determined, and we can adjust the elevation angle of our artillery according to Example 3 to fight back and destroy the other side.
This example can't be solved without using the equation method. Equation method only needs to be considered step by step, each step is very simple, and it does not need deep thinking or high IQ. Everyone can do it, especially when calculating, which is completely fixed and can be done by computer.
The human brain is powerful, but its defects are obvious, its memory is limited, it can't reason remotely, its concept is changeable, and it can't consider multiple factors at the same time. Mathematical tools can just overcome these defects, replacing quantitative or extremely abstract concepts with symbols, so as to ensure that the connotation and extension remain unchanged in the process of reasoning. Find the relationship equation in pairs, then transform the equation according to only a few calculus rules, and finally get the exact value of the unknown quantity. This reasoning method does not need memory or brain, it can be calculated on paper, and everyone can learn it. With the development of information technology, the process of mathematical calculus has been solved by all kinds of excellent software, which further lightens the burden on the human brain and can be solved by inputting the quantitative relationship between factors into the computer.
It can be said that the development of science depends entirely on mathematical reasoning tools. Only by mastering basic mathematical tools can modern people understand science and technology. Especially for complex problems, the relationship equation is often the relationship between the rates of change, that is, the differential equation. Reasoning is completely mathematical calculus, and understanding becomes irrelevant to intuition, and can only be understood from the rules of mathematical calculus. If it is a multivariate partial differential equation, the physical vector represented by complex numbers is especially so in understanding.