Addition, subtraction, multiplication and division. The sum of two complex numbers is still a complex number, its real part is the sum of the original two complex numbers, and its imaginary part is the sum of the original two imaginary parts. The addition of complex numbers satisfies the commutative law and associative law. In addition, when complex numbers are used as the base, exponent and real number of power and logarithm, their operation rules can be derived from Euler formula E I θ = COS θ+ISIN θ (arc system).

Numbers in the form of a+bi (both a and b are real numbers) are complex numbers, where a is called the real part, b is called the imaginary part, and I is the imaginary part. Complex numbers are usually represented by z, that is, z=a+bi. When the imaginary part b of z = 0, z is a real number. When the imaginary part b≠0 and the real part a = 0 of z, z is often called pure imaginary number.

Complex number field is an algebraic closure of real number field, that is, any polynomial with complex coefficients always has roots in complex number field. The complex number was first put forward by Cardan, a scholar in Milan, Italy, in the16th century. Through the work of D'Alembert, De Moivre, Euler and Gauss, this concept was gradually accepted by mathematicians.

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."

However, truth can stand the test of time and space and finally occupies its own place. The French mathematician D'Alembert (17 17 ~ 1783) pointed out in 1747 that if the imaginary number is operated according to the four algorithms of polynomials, its result is always in the form of a+bi (both a and b are real numbers).

French mathematician De moivre (1667 ~ 1754) discovered the famous democritus theorem in 1722. Euler found the famous relation in 1748, and expressed the square root of-1 for the first time in the article Differential Formula (1777). He creatively used the symbol I as the unit of imaginary number.

The imaginary number is not imaginary, but it does exist. Norwegian surveyor wessel (1745 ~ 18 18) tried to give an intuitive geometric explanation of this imaginary number in 1797, and published his practice for the first time, but it did not get the attention of academic circles.