We call a number in the form of z = a+bi (both a and b are real numbers) a complex number, where a is called the real part, b is called the imaginary part, and I is called the imaginary part. When the imaginary part of z is equal to zero, z is often called a real number; When the imaginary part of z is not equal to zero and the real part is equal to zero, 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.

Definition and representation of imaginary part

definition

Complex number z=x+iy, where x and y are arbitrary real numbers, x is called the real part of complex number z, and y is called the imaginary part of complex number z? [1] (note that the imaginary part does not include the imaginary unit I)

Algebraic representation

In English, Real numbers aRe real quantities, so the first two letters "re" of real are generally taken to represent the real part of complex numbers; The Imaginary number is an imaginary number, so the first two letters "im" of the imaginary number are generally taken to represent the imaginary part of the complex number. For example:

Re(2+3i)=2，Im(2+3i)= 3；

Re(-7.38i)=0，Im(-7.38 I)= 7.38。

Complex plane representation

The point (x, y) on the complex plane represents the complex number x+iy, where the y axis is the imaginary axis and the value of y is the imaginary part.

A function that defines the real and imaginary parts of a complex number

First, it is stipulated that two complex numbers are equal.

We stipulate that two complex numbers are equal if and only if their real and imaginary parts are equal respectively.

From the vector point of view, because a 1=a2 and b 1=b2, the complex number a 1+b 1i has the same modulus as the two vectors represented by the complex number a2+b2i, and the two vectors have the same direction.

Second, the definition of * * * yoke complex number

When the real parts of two complex numbers are equal and the imaginary parts are opposite, these two complex numbers are called * * * yoke complex numbers.

Complex numbers a+bi and a-bi are complex numbers of * * * yoke.

A+bi times a-bi equals a2+b2.

Third, define the module of complex number.

Using Pythagorean theorem, we can find the distance from the point representing complex number to the origin on the complex plane.

Fourth, define the principal values of complex numbers.