one
Definition of equality of two complex numbers:
If the real and imaginary parts of two complex numbers are equal, then we say that these two complex numbers are equal, that is, if A, B, C and d∈R, then a+bi=c+di.
A=c, B = D. Especially, when A, b∈R, a+bi=0.
a=0，b=0。
The necessary and sufficient conditions for the equality of complex numbers provide a way to turn complex problems into practical problems.
Special reminder of plural equality:
Generally speaking, two complex numbers can only be said to be equal or unequal, but their sizes cannot be compared. If both complex numbers are real numbers, the sizes can be compared, and only if both complex numbers are real numbers can the sizes be compared.
The method steps to solve the complex equation problem:
(1) Converts a given complex number into the standard form of a complex number;
(2) Solving complex numbers according to necessary and sufficient conditions.
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The concept of complex number:
A number in the form of a+bi(a, b∈R) is called a complex number, where I is called an imaginary unit. The set formed by all complex numbers is called a complex set, which is represented by the letter C.
Representation of complex numbers:
Complex numbers are usually represented by the letter z, that is, z=a+bi(a, b∈R). This representation is called algebraic form of complex number, where A is the real part of complex number and B is the imaginary part of complex number.
Geometric meaning of complex numbers:
(1) Complex plane, real axis and imaginary axis:
The abscissa of point z is a and the ordinate is b, and the complex number z=a+bi(a, b∈R) can be represented by point Z(a, b). The plane that establishes a rectangular coordinate system to represent a complex number is called a complex plane, the X axis is called a real axis, and the Y axis is called an imaginary axis. Obviously, all points on the real axis represent real numbers, and all points on the imaginary axis represent pure imaginary numbers except the origin.
(2) Geometric meaning of complex number: the set of complex number set C has a one-to-one correspondence with all points on the complex plane, namely
This is because every complex number has a unique point on the corresponding complex plane; On the contrary, every point on the complex plane has a unique complex number corresponding to it.
This is a geometric meaning of complex number, and it is another representation of complex number, that is, geometric representation.
Modules of complex numbers:
The distance from the point Z(a, b) corresponding to the complex number z=a+bi(a, b∈R) on the complex plane to the origin is called the module of the complex number, which is denoted as |Z|, that is |Z|=
Imaginary unit I:
(1) Its square is equal to-1, that is, I2 =-1;
(2) Real numbers can be used to perform four operations, and the original laws of addition and multiplication are still valid when performing four operations.
(3) Relationship between I and-1: I is the square root of-1, that is, one root of equation x2=- 1, and the other root of equation x2=- 1 is-i.
(4) periodicity of I: i4n+ 1=i, i4n+2=- 1, i4n+3=-i, i4n= 1.
Properties of complex modulus:
The relationship between complex number and real number, imaginary number, pure imaginary number and 0;
For the complex number a+bi(a, b∈R), if and only if b=0, the complex number a+bi(a, b∈R) is a real number a; When b≠0, the complex number z=a+bi is called an imaginary number; When a=0 and b≠0, z=bi is called pure imaginary number; Z is a real number 0 if and only if a=b=0.
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