Mathematics has many objects and structures, what do we want to do with them. For example, given a number, we will double it according to the context, square it, or count it down; given an appropriate function , we may want to differentiate it; given a geometric figure, we may want to transform it, etc.
If we define a mathematical program, then inventing the technique of executing this program is an obvious mathematical plan. This leads to the so-called direct question about this program . However, there is also a more profound so-called anti-problem , which is in the form as follows. Assuming that the answers obtained by the program and the execution of the program are given, can we figure out what mathematical object the program is used on? will be very clear if I tell you that there is a number, square it and the result is 9. Can you tell me what this number is? It's very simple, the answer is 3 or -3.
If you want to discuss this issue in a more formal way, you will say that you were just studying the equation x^2=9, and you found that it has two solutions. Such an example raises three general questions: Is there a solution to an equation of
- ? If
- , does it happen to be a solution? What kind of sets do these solutions in
- ? The first two problems of
are called the existence and uniqueness of solutions. The third problem does not have much significance in the case of equation x^2=9, but in more complex cases, such as partial differential equations, it may be a very important problem.
In a more abstract language, let f is a function, and there is such a proposition in front of it, its form is f(x)=y. The direct problem is to find y given x, and the inverse problem is to find x given y. This inverse problem is to solve the equation f(x)=y. The problem of solving this form of equation is closely related to the reversibility problem of function f . Because x and y may be objects that are much more general than numbers, the concept of solving equations itself is very general, so it is one of the central problems of mathematics.
linear equation
The equation that primary school students first encountered is typically an equation like 2x+3=17. To solve such a simple equation, we regard x as an unknown number, and unknown numbers must also obey the usual rules of arithmetic. Using these laws, you can turn this equation into a much simpler equation: subtract 3 from both sides of the equation and get 2x=14, and then use 2 to divide both sides of the new equation and get x=7. We actually proved that if there is a certain number x such that 2x+3=17, then this number must be 7. What we have not proved yet is that there is indeed such a number x. So, strictly speaking, there should be another step, namely, verification 2×7+3=17. Here, it is obviously correct, but for more complex equations, the corresponding conclusions may not always be correct, so this last step is still important.
equation 2x+3=17 is called linear equation . This is because the function acting on x (multiply by 2 and then add 3) is a linear function. As I just saw, a linear equation containing only one unknown number is easy to solve, but if you want to solve an equation with more than one unknown number, the situation will be more complicated. Consider a typical example of an equation containing two unknown numbers, that is, equation 3x+2y=14. There are many solutions to this equation. After selecting a y, you can make
so there is a pair (x, g) to satisfy this equation. To make the problem a little harder, you can add another equation, such as 5x+3y=22, and then try to solve these two equations at the same time. The result at this time is that there is only one (a set) solution x=2 and y=4. Generally speaking, two linear equations containing two unknown numbers happen to have a set of solutions. If you look at this situation from the perspective of geometry, it is easy to understand. An equation shaped like ax+by=c is an equation of a straight line on the xy plane. Two straight lines intersect normally at one point, the exception is that the two straight lines are the same. At this time, they intersect at infinite points, or they are parallel, and at this time they do not intersect at all.
If there are several equations containing several unknown numbers, consider them as an equation with that contains an unknown thing , it will be simpler in concept.This sounds completely impossible, but it is entirely possible to allow this unknown thing to be a more complex object. For example, the two equations of 3x+2y=14 and 5x+3y=22 can be written as a single equation containing matrix and vector
If A represents the above matrix, x represents an unknown vector, and b represents a known vector, then this equation becomes Ax=b, which looks much simpler, although in fact it only hides the complexity in the symbol.
However, this process is not just "sweeping the garbage under the carpet and hiding it", but there are more things. On the one hand, simple symbols cover up many specific details of this problem, but on the other hand, they also reveal something that cannot be seen: there is a linear mapping from R^2 to R^2. What I want to know is which vector x is mapped as vector b (if there is such a vector). If we encounter a specific system of simultaneous equations, this is not too different, and the calculations we need to do are still the same. But if you want to make general reasoning, a matrix equation containing a single unknown vector is much easier to consider than a unified system of equations containing several unknown numbers. This phenomenon will occur throughout mathematics and is the main method to study high-dimensional space .
polynomial equation
We just discussed the generalization of linear equation from one unknown number to multiple unknown numbers. Another direction to generalize them is to treat linear equations as 1-order polynomials, while considering functions with higher numbers of degrees. For example, in middle school, we have learned how to solve quadratic equations such as
. A more general polynomial equation is like
to solve such an equation, which is to find the value of x to satisfy this equation. This seems to be an obvious thing, but it is not so obvious when encountering an equation like
. The solution to this equation is
So, what is root number 2? Its definition is a positive number equal to 2 after square. However, it seems that this equation has not been "solved" yet by saying that x is equal to positive or negative and is squared to 2. Even saying x=1.4142135... is not completely satisfactory, because this is just writing a short paragraph of an endless formula, and it is impossible to see any pattern in this formula that can be distinguished.
From this example, you can get two introductions:
- . One is that for an equation, the important thing is often the existence and properties of the solution, rather than whether the formula for the solution can be found.
Although we say
doesn't let us learn anything, this conclusion does contain a fact: 2 has square root . This is usually proposed as an inference of the so-called intermediate value theorem . This theorem states that if f is a continuous real-valued function, and f(a) and f(b) are each on the side of zero, then somewhere between a and b, there must be a real number c such that f(c) = 0. This result can be used for the function f(x)=x^2-2, because f(1)=-1, and f(2)=2. So there must be an x between 1 and 2 such that x^2-2=0. For many purposes, it is sufficient to know the existence of this x, plus knowing that the properties of this x are defined so that it is positive and squared later to 2.
Use a similar argument to know that all positive real numbers have positive square roots. But when we try to solve more complex quadratic equations, the situation is different. There are two ways to choose from at this time. For example, considering the equation
, we will notice that when x=4, its value is -1, and when x=5, its value is 2. From the intermediate value theorem, we know that this equation has a solution between 4 and 5. However, if you use the matching method,
will get two solutions, which is more information than using the intermediate value theorem. We have proved that root number 2 exists and know that its values are between 1 and 2.Now we not only know that the equation x^2-6x+7=0 has a solution between 4 and 5, but also know that this solution is closely related to the solution of equation x^2=2. It can even be said that this solution is constructed from the deconstruction of equation x^2=2.
This proves that there is a second important aspect of solving equations, that is, in many cases, the explicit solatability of the solution is a relative concept of . As long as you give a solution to the equation x^2=2, when solving the more complex equation x^2-6x+7=0, you no longer need to get any new input from the intermediate value theorem, and only a little algebra is needed.
However, the root number 2 in this expression is not defined by an explicit formula, but is defined as an real number . This real number has some properties, and we can prove that it exists.
solves higher order polynomial equations than solves quadratic equations, and thus creates many attractive problems. In particular, solving 3 or quadratic equations has complex formulas, but solving 5 and higher order equations has been a well-known unsolved problem for hundreds of years. It was not until the 19th century that Abel and Galois proved that the formula for explicit solutions was not found.
polynomial equation
has such an equation
We can see that it has many solutions: if x and y are fixed, a cubic polynomial equation of z is obtained. All cubic polynomial equations have (at least one) real solutions, so for each fixed x and y, there is a certain z that makes triple (x, y, z) the solution to this equation.
Because the formula of solutions to cubic equations is very complex, it doesn't make sense to accurately describe the set of all these triplets (x, y, z). However, if you regard this set of solutions as a geometric object, is a 2-dimensional surface in space, and consider some qualitative questions about it, you can learn a lot from it. For example, we may want to understand its general nature. In the language of topology , we can explain these issues clearly.
can of course be further generalized to consider the simultaneous solution of several polynomial equations. Understand the solution sets of these systems of equations belong to the field of algebraic geometry .
Dipantu equation
Whether there is a solution for a specific equation depends on where you are allowed to solve it. If only x is allowed to be a real number, then there is no solution to the equation x^x+3=0, but in complex numbers, it has two solutions. There are infinite solutions to the equation x^2+y^2=11, but if you find that x and y are both integers, there is no solution to this equation.
The above example is a typical Diopian equation. Whenever you see this noun, you will indicate that you require its integer solution . The most famous Diopantu equation is the Fermat equation
Thanks to Wiles' work, it is now known that when n is greater than 2, it has no positive integer solution. In contrast, the equation x^2+y^2=z^2 has infinitely many integer solutions. A large part of the modern algebraic number theory is directly or indirectly discussing the Diophant equation. Just as with equations of real or complex numbers, it is fruitful to discuss the structure of sets of solutions to Diopani equations, and this type of research belongs to the field of arithmetic geometry . A notable feature of the
Diopiant equation is that they are extremely difficult. So naturally, it would be doubtful whether there might be a system for them to handle , the 10th question in the list of famous questions raised by Hilbert in 1900. But it was not until 1970 that Yuri Matiyasevich pointed out that the answer to this question was no. An important step in solving the problem of
was made in 1936 by Church and Turing . This step was only taken after formalizing the concept of algorithm (in two different methods) and thus clarifying the concept of " systematically processing ".Before the computer age, this was not easy, but now we can recapitulate the solution of Hilbert's 10th problem as follows:
Want to find an computer program so that after inputting any Diopani equation, if there is a solution for this equation, it will definitely output "YES", and when there is no solution, it will definitely output "NO", and never make any mistakes, which cannot be done.
What does this tell us about the Diopantu equation? We can no longer dream of a final theory that encompasses all such equations, instead we are forced to focus on the special categories of such equations and develop different solutions to them. If it weren't for the notable connection between the Diophant equation and the very general equations of other parts of mathematics, this would seem to make the Diophant equation uninteresting after solving the first few equations.
For example, the equation
looks very special, but in fact, the elliptic curve it defines is the central problem of modern number theory (including the proof of Fermat's theorem ). Of course, Fermat's theorem itself is also a Diophant's equation, but its research has led to major developments in other parts of 's number theory . The correct conclusion that should be drawn may be: if the result of solving a special Diophant equation is not just to add one to the already solved list of equations, then it is attractive and worth studying.
differential equation
So far, the equations we consider are all unknowns with numbers or points in n-dimensional space . To generate such equations, we do different combinations of basic operations of arithmetic and then apply them to something unknown.
Two famous differential equations are given below to compare with the equations discussed in the past:
The first is "normal" differential equation , which is a simple harmonic equation of motion, which has the general solution
The second is "partial" differential equation , which is a thermal equation.
has many reasons to show that solving differential equations is a leap in precision.
- One reason is that what is unknown to now is the function , which is much more complex than points in a number or n-dimensional space.
- The second reason is that the operational differentials and integrals applied to the function are far less "basic" than addition and multiplication. The three reasons for
- are that differential equations, even if they are natural and important equations, can be solved in "closed form", that is, a formula is used to represent the unknown function f, but it is an exception and unconventional.
Now go back to the first equation,
This means that the differential equation can be regarded as a case where a matrix equation is generalized to infinite multi-dimensional. Thermal equation also has the same properties: if Ψ (T) is defined as
then Ψ is another linear map. This differential equation is called linear, and their obvious connection with linear algebra makes them much easier to solve. A useful tool in this regard is Fourier transform .
What about those more typical equations, that is, equations that cannot be solved in closed form? At that time, the focus once again shifted to . Is there a solution in ? If so, what are their properties? Like polynomial equations, this depends on what is considered a permissible solution. Sometimes, we are like studying the equation x^2=2 again: it is not difficult to prove that the existence of a solution is just a name. The equation
is a simple example. In a sense, it cannot be solved. It can be proved that a "basic" function constructed from polynomials, exponential functions , trigonometric functions , etc., and after differentiation, you will get e^ (-x²). However, in another sense, this equation is easy to solve. You only need to integrate the function e^ (-x²), and the resulting function is a normal distribution function.This function has basic importance in probability theory, so it is given a name (mark):
In most cases, it is hopeless to write the formula of the solution. A famous example is the three-body problem: give three moving objects in space ( particle ), and let them attract each other with gravity, ask how they will continue to move? Newton's law can be used to write differential equations that describe this situation. For two moving objects, Newton solves the corresponding equations and thus explains why the planets move along an elliptical orbit around the sun, but for three or more objects, these differential equations have proved to be very difficult to solve. It is now known that there are profound reasons for this difficult situation: at this time, these differential equations will lead to chaotic state . However, this opens the way to study very interesting questions such as chaos and stability.
Sometimes, there are ways to prove that solutions exist, even if these solutions cannot be easily determined. At this time, it is not necessary to obtain an accurate formula, but only a general description is desired. For example, if this equation has time dependence (for example, thermal equations and wave equations are both), people will ask whether the solution decays, blasts, or generally remains unchanged with time? These more qualitative questions are called asymptotic problems , and there are some tips to answer some questions in this category, although there is no explicit formula to give the solution. The same situation in
is the case with the Diophant equation. Partial differential equations include some special and important classes in nonlinear partial differential equations, which can accurately write the solution. This gives a very different style of research: people once again focus on the properties of solutions, but this time it is a more algebraic property in nature, that is, the formula of solutions will play a more important role.
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