The terms "complex number" and "imaginary number" are both introduced when people solve equations. In order to find the roots of quadratic and cubic equations in one variable, we will encounter the problem of finding the square root of negative numbers. 1545, the Italian mathematician cardano (150 1 ~ 1576) studied imaginary numbers for the first time in his book Great Skills and made some calculations. 1572, the Italian mathematician rafaclbombclli (1525 ~1650) formally used the terms "real number" and "imaginary number". Since then, German mathematician Leibniz (1646 ~ 17 16), Swiss mathematician Euler (1707 ~ 1783) and French mathematician abramov. 1667 ~ 1754) and others also studied the relationship between imaginary numbers and logarithmic functions and trigonometric functions. In addition to understanding the equation, they also used it in calculus and other aspects, and got many valuable results, which made some complicated mathematical problems simple and easy. Around 1777, Euler expressed the square root of-1 for the first time, 1832, and the German mathematician Gauss (1777 ~ 1855) introduced the concept of complex number for the first time. Complex numbers can be represented by +BI. Gauss also corresponds complex numbers to points on the complex plane one by one, and gives the geometric explanation of complex numbers. Soon, people linked complex numbers with plane vectors and made them widely used in electrotechnics, fluid mechanics, vibration theory and wing theory. Then the theory of "complex variable function" with complex number as variable is established, which is a brand-new and powerful branch of mathematics, so we should deeply understand the truth that "imaginary number is not empty"

16th century Italian Milan scholar Jerome Cardan (1501-1576) 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 as a formula. When discussing whether it is possible to divide 10 into two parts to make their product equal to 40, he wrote the answer as =40. Although he thought the expressions sum were meaningless, fictitious and illusory, he divided 10 into two parts and made their product equal to 40. 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 Leibniz (1646-16) said in 1702: "imaginary number is a subtle and strange hiding place for gods, and it is probably an amphibian in the field of existence and falsehood." Swiss mathematician Euler (1707- 1783) said; "In all forms, it is impossible to learn mathematics. Imagine numbers, because they represent the square root of a negative number. For such figures, we can only assert that they are neither nothing nor more than nothing, nor less than nothing. They are purely illusory. " However, truth can stand the test of time and space and finally occupies its own place. The French mathematician D'Alembert (1717-1783) pointed out in 1747 that if the imaginary number is operated according to the four algorithms of polynomials, its result will always be in the form of (A and B are both real numbers) (Note: the symbol =-is not used in the current textbooks. The French mathematician Demofer (1667- 1754) discovered this formula in 1730, which is the famous Demofer theorem. 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. "Imaginary number" is not an imaginary number, but it does exist. 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.

German mathematician Gauss (1777- 1855) published the graphic representation of imaginary numbers in 1806, that is, all real numbers can be represented by a number axis, and similarly, imaginary numbers can also be represented by points on a plane. In the rectangular coordinate system, take the point A corresponding to the real number A on the horizontal axis and the point B corresponding to the real number B on the vertical axis, and draw a straight line parallel to the coordinate axis through these two points, and their intersection point C represents the complex number A+Bi. 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.

After many mathematicians' unremitting efforts for a long time, the complex number theory has been deeply discussed and developed, which makes the ghost of imaginary number, which has been wandering in the field of mathematics for 200 years, unveil its mysterious veil and reveal its true colors. Original imaginary number is not empty. The imaginary number has become a member of the family of number systems, so the real number set has been extended to the complex number set.