Several common problems of complex numbers

Lu Cailing, Qing Ji City, Shandong Province

First, use the algebraic form of complex numbers.

From the algebraic form of complex numbers, method of substitution is the most basic and commonly used method to solve problems.

Example 1 It is known that if, the maximum value of is ().

a . 6b . 5c . 4d . 3

Analysis: Suppose, then.

,,,

.

,,, so choose C.

Second, the necessary and sufficient conditions for using complex numbers to be equal

In a complex set, take two numbers,, and at will.

Example 2 The complex number is known, and the real number is used.

Solution:

.

Because they are all real numbers, they pass.

These two types add up to form.

Solve,

Corresponding,

So, the real number is, or ...

Thirdly, by using the division rule of complex numbers and the operational properties of imaginary numbers

1. can be multiplied by the * * * yoke complex number of the denominator to make the denominator "real";

2. Remember some common results:

(1) periodicity;

(2);

(3),;

(4);

(5) set, the attribute is:

①;

②,;

③.

Assume that in Example 3, the number of elements in the collection is ()

A.1b.2c.3d. Infinitely many

Analysis: Because,

So when,,,,,

Yes, so the answer is C.

Fourth, use plural yokes.

Complex numbers and complex numbers are complex numbers that are yokes to each other.

Example 4 If the equation is the root, evaluate it.

Solution: Because it is a real number, the sum of the two is a real number, and the product of the two is a real number;

Because it is a root of the equation, the other root that meets the conditions must be its * * * yoke complex number, so solve it.

Another solution: substitute into the equation and get the sum according to the necessary and sufficient conditions for the equality of complex numbers.

Note: the product of two * * * yoke complex numbers is:, that is, the product of two * * * yoke complex numbers is equal to the square of complex modulus.

Example 5 If,, then ()

A. Pure imaginary number B. Real number C. Imaginary number D. Uncertainty

Analysis: A number is itself a real number if its * * * yoke complex number is.

From, we can know that it is a real number.

So the answer is B.

Fifth, using the geometric meaning of complex numbers

1. Use the modulus of a complex number

Modules of complex numbers.

Example 6 summarizes the requirements.

Solution:.

Note: If you simplify first and then find the module, the amount of calculation will increase.

2. Using the geometric meaning of complex addition and subtraction

The addition (subtraction) method of complex numbers can be operated according to the parallelogram (triangle) rule of vectors.

Example 7 Set a complex number, satisfy and seek.

Solution: Draw a parallelogram as shown in the figure according to the meaning of the question.

So ...

Therefore, ...

Yes

We see that the above problem-solving methods are interrelated, so we should pay attention to solving problems flexibly and comprehensively apply what we have learned. From/View /9p4odu.