As early as a week ago, I wrote a paper on the existence of imaginary parts in complex numbers. Because time is tight, it is meaningless, quite shallow and even has many unscientific loopholes.

The previous understanding of virtual field was completely based on a empty word. Because of the meaning of complex variables, the book gives: because of the need to solve algebraic equations, people lead to complex numbers.

With the appearance of complex numbers, the square root operation in basic operations has no solution, and this polynomial has no root increase, which provides help for human operations in some logical fields.

In order to illustrate the exploration of two kinds of cognition, the following is part of what I discussed in my previous paper (this part is discussed under the cognition that I think imaginary numbers are completely fictitious):

"The set of complex numbers-the complex plane is a two-dimensional plane, but it is not any two-dimensional plane in our three-dimensional world. It can be said that the complex plane can not find a specific one-to-one correspondence in the real world at all, and it is a purely created two-dimensional plane. I am curious about the abstraction of this idea, so I hope to find a positive solution.

Just recently, I clarified two concepts through a forum debate: mathematics and science. The conclusion is that mathematics is not science. Mathematics does not belong to the category of science, but a kind of logic, a discipline as a tool; Science is a collection of theories. Even a false proposition, such as geocentric theory, is science. To distinguish whether a subject is scientific or not, we need another subject as its judgment basis: falsificationism. Finally, I was convinced by a theory put forward by Jie Bing: what can be proved or falsified belongs to science; And mathematics cannot be falsified.

This shows to some extent that mathematics is a metaphysical subject, even including geometry. In mathematics, in my opinion, the metaphysical interest in the field of complex numbers is more prominent.

I have seen some people take imaginary numbers and quantum theory as examples when discussing metaphysics. I once thought that the unknowns without observers in quantum theory could be represented by imaginary numbers. Of course, now it seems that this is a very shallow idea. It's like mapping the famous paradox-Schrodinger's cat's life and death to the complex number field. When I was in high school, I made a very shallow and unscientific proof of similar metaphysics. If the cat's life and death, that is, whether uranium decays or not, is mapped to a complex number field, then in order to correspond to the uniform distribution of uranium decay probability, we might as well map it to a group of * * * yoke complex numbers. When the observer appears, the cat's life and death are determined, and the uncertainty disappears, then the nonexistence of the complex number it maps should also disappear, that is, the complex number is embodied in the real number field, and the corresponding operation is modular, which shows that the modulus of the * * * yoke complex number is equal, which is contradictory to the difference between the cat's life and death after determination. Of course, this simple reasoning itself is not very scientific. But the conclusion should be a positive solution: uncertainty does not mean non-existence, and the two cannot be mapped to each other.

This at least shows that the existence of complex numbers may be isolated in disciplines outside the field of mathematics. The complete metaphysics of the world outlook is unrealistic. "

Above.

After the completion of this naive paper, I feel that I don't have any in-depth knowledge understanding of the existence meaning of complex plane and imaginary number, just some personal thoughts, which are quite inappropriate. In order to understand this problem more accurately and scientifically, I consulted some related materials.

First of all, the history of imaginary numbers is like this:

Pt 1。

Cardan (1501-1576), an Italian Milan scholar in the 6th century, published the general solution of cubic equation in his book "Important Art" in 1545, which was later called "Cardan Formula". He was the first mathematician to write the square root of a negative number into a formula. The French mathematician Descartes (1596-1650) gave the name "imaginary number", and he made it correspond to "real number" in geometry (published in 1637). Since then, imaginary numbers have spread. A new star, imaginary number, was found in the number system, which caused a chaos in mathematics. Many great mathematicians do not admit imaginary numbers. German mathematician Karl Zebnitz (1664- 17 16) said in 1702: "The imaginary number is a subtle and strange hiding place of God, and it is probably an amphibian in the realm of existence and falsehood." Swiss mathematician Euler (1707- 1783) said; "All shapes and numbers are impossible, imaginary numbers, because they represent the square root of negative numbers. For such figures, we can only assert that they are purely false. " However, truth can stand the test of time and space and finally occupies its own place. The French mathematician Da Lambert (1717-1783) pointed out in 1747 that if the imaginary number is operated according to the four algorithms of polynomials, then its result is always in the form of (A and B are both real numbers). The French mathematician Dimover (1667-1754) discovered the famous Tamover theorem in 1730. Euler found the famous relation in 1748. In his article Differential Formula (1777), he first expressed the square root of 1 with I, and he pioneered the use of the symbol I as the unit of imaginary number. 1745- 18 18, a Norwegian surveyor tried to give an intuitive geometric explanation of the imaginary number 1779, and published his practice for the first time, but it did not get the attention of academic circles. The German mathematician Gauss (1777-1855) published an image representation of imaginary numbers in 1806. In this way, the plane whose points correspond to complex numbers is called "complex plane", and later it is also called "Gaussian plane". 183 1 year, Gauss expressed the complex number A+Bi with real array (a, b), and established some operations of complex numbers, making some operations of complex numbers "algebraic" like real numbers. He first put forward the term "complex number" in 1832, and also integrated two different representations of the same point on the plane-rectangular coordinate method and polar coordinate method. Unifying the algebraic form and triangular form representing the same complex number, the points on the number axis correspond to the real number-1, and the points extended to the plane correspond to the complex number-1. Gauss regarded the complex number not only as a point on the plane, but also as a vector, and expounded the geometric addition and multiplication of the complex number by using the corresponding relationship between the complex number and the vector. At this point, the complex number theory has been established completely and systematically.

Pt 2。

"Imaginary number" is an imaginary number invented by human beings in the process of developing mathematical problem-solving technology. It does not exist in real life, and it is not needed in practical business mathematics. "Complex number" can solve some physical and mathematical problems, and the real number solution obtained by the final transformation will have physical significance, while the complex number with imaginary number will be meaningless at that time.

So far, the theory that imaginary numbers do not exist in physics is still correct in my understanding. Space vector algebra to see the law of time;

Time has spatial directionality and can be used as vector algebra. When we used to do algebraic operations, imaginary numbers were time. Doppler effect is one of the experimental foundations to prove the existence of four-dimensional time.

Imaginary number does not exist in the three-dimensional world, but it is defined as the fourth-dimensional time. Virtual time is just a way of mathematical presentation and a way of processing. Just like RCL circuit, we also use imaginary number to deal with the phase angle relationship, but the inductance itself is not imaginary. This is an artificial definition, but it also reveals some physical characteristics of imaginary numbers in a certain sense.

Then I got the concept of superluminal particles in physics: superluminal particles are theoretically predicted particles. It has a local velocity (instantaneous velocity) that exceeds the speed of light.

Its mass is virtual, but its energy and momentum are real.

Some people think that such particles cannot be detected, but this is not necessarily the case. Examples of shadows and light spots show that things faster than the speed of light can also be observed.

At present, there is no experimental evidence of the existence of superluminal particles, and most people doubt its existence. It is claimed that in the experiment of measuring the mass of neutrinos released by the beta decay of tritium, there is evidence that these neutrinos are tachyons. This is doubtful, but this possibility cannot be completely ruled out.

Although the tachyon has not been recognized by the scientific community, at least human beings have applied imaginary numbers to physics. Once proved, the idea that imaginary numbers have no physical meaning is broken.

This is undoubtedly two in-depth explorations of the meaning of imaginary numbers!

I think the following passage objectively and positively illustrates the practical significance of imaginary numbers:

"The main task of algebra is to give as many answers as possible to this question. By introducing imaginary numbers, those' meaningless' roots are not a problem at all. However, in history, the existence and significance of imaginary numbers have caused heated debates. Imaginary numbers are ridiculed as' ghosts of numbers', and some great mathematicians like Descartes refuse to admit this. This controversy did not subside until 1800 when imaginary numbers were successfully explained by geometry. For pragmatists, imaginary number is of course a calculation tool, as long as it is useful, but not for serious mathematicians. Gauss once said that the key is not application, but that if we discriminate against these imaginary numbers, the whole analysis will lose a lot of beauty and flexibility. Why do you think "discrimination against imaginary numbers" is not beautiful? I think this is because the second law of mathematical beauty is at work: the law of symmetry. When we regard imaginary number and real number as the same truth, but they belong to the horizontal axis and vertical axis of a unified complex plane respectively, the solutions of all algebraic equations are symmetrical for real number and imaginary number. And any artificial' discrimination' will break this symmetry. "

Through the study of the course, we can know that complex numbers can be applied to mathematical modeling in reality and have incredible properties and laws in many operations. The introduction of complex numbers provides many new ways for people to solve the real number field and physical science, and opens many roads that could not be unblocked. Both magic residue and conformal mapping provide new ideas and convenience for people to solve problems in non-complex fields.

Imaginary number, whether it exists objectively or not, is beautiful!

My opinion, please sort it out again I also want to write a complicated paper, but I also want to write a paper on integral transformation.