Teaching objectives

Knowledge and skills

1, understand the process of number system expansion and the necessity of introducing complex numbers.

2. Master the related concepts and algebraic symbol forms of complex numbers, the classification methods of complex numbers and the necessary and sufficient conditions for the equality of complex numbers.

Process and method

1, through the introduction of number series expansion, let students understand the general law of number series expansion.

2. Through the process from concrete to abstract, let students form the general form of complex numbers.

Emotional attitudes and values

1, experience the innovative spirit and practical spirit contained in the process of expanding the number system, and feel the role of human rational thinking.

2. Mathematical thinking method of experiencing analogy, classified discussion and equivalent transformation.

Emphasis and difficulty in teaching

Emphasis: the necessity of introducing complex numbers and related complex numbers, the classification of complex numbers, and the necessary and sufficient conditions for the equality of complex numbers.

Difficulties: the introduction of imaginary unit I and the concept of complex number

teaching process

(A) the introduction of the problem

In fact, x and y really don't exist in the real number range? Why is this happening? Assuming that x and y exist, they must be some numbers that are not real numbers. So, what are these numbers? Can we solve this problem? This is what we are going to learn today, the expansion of the number system and the introduction of complex numbers.

(b) Review the expansion of the digital system.

Teacher: Actually, for this kind? Is there not enough quantity? We are no strangers to this situation. Do you remember? From primary school to now, we have been experiencing the continuous expansion of numbers. Now let's go back and see how it was solved before. Is there not enough quantity? The problem.

(3) Analogy, introducing new numbers and expanding the set of real numbers.

1, simulate the expansion law of number system and guide students to find a solution? Not enough real numbers? The solution to this problem

Health: Introduce a new number to make the square negative.

Teacher: We hope that the square of the number to be introduced is negative, but there are infinitely many negative numbers. We refuse to introduce so much at once, just introduce the square.

2. Historical representation:

3. Explore the general form of complex numbers:

(d) duplicate of the new number set

Definition of 1. Complex number (omitted)

2. Application of complex numbers: Complex numbers are widely used in mathematics, mechanics, electricity and other disciplines. Complex numbers are closely related to vectors, plane analytic geometry and trigonometric functions. , and is the basis for further study of mathematics.

(5) Classification of complex numbers

(6) Necessary and sufficient conditions for the equality of complex numbers

The necessary and sufficient condition of complex equation can transform the problem of complex equation into the problem of solving equation, which is a transformation idea.

Summary after class

1. Due to practical needs, we summarized the law of the triple expansion process of numbers. By analogy, we introduce a new number I, extend the real number set to the complex number set, understand the algebraic form of complex numbers, discuss the classification of complex numbers and the necessary and sufficient conditions for the equality of complex numbers, and transform the complex number problem into the solution of equations by using the equality conditions.

2. So, what exactly is a complex number? Can you find its shadow in reality like a real number? Don't worry, our exploration will not stop. This is what we will discuss next time.

homework

1, exercise 3. 1 group 1 1 and 2.

2. Can you compare the sizes of complex numbers after class? Why? (available information)

High school mathematics elective course teaching plan 2 1-2 "the promotion of number system and the concept of complex number" learning objectives;

1, understand the necessity of introducing complex numbers; Understand and master the unit I of imaginary number

2. Understand and master the laws of the four operations of imaginary and real numbers.

3. Understand and master the related concepts of complex numbers (complex set, algebraic form, imaginary number, pure imaginary number, real part and imaginary part) Understand and master the related concepts of complex equation.

Learning focus:

The concept of complex number, imaginary unit I, the classification of complex number (real number, imaginary number, pure imaginary number) and complex phase are the teaching focuses of this lesson.

Learning difficulties:

Autonomous learning

First, knowledge review:

The concept of number came into being and developed from practice. Due to the need of counting, 1, 2 and its representation are produced. No? The number of 0. The whole composition of natural numbers natural number set n In order to solve the problem of equal division of some quantities in measurement and distribution, people introduced fractions; In order to express various quantities with opposite meanings and meet the needs of counting, people introduced negative numbers. Thus, the number set is extended to the rational number set Q. Obviously, n Q. If natural number set (including positive integers and 0) and negative integer sets are combined into an integer set Z, then there are Z Q and N Z. If the integer is regarded as a fraction with the denominator of 1, then the rational number set is actually a fraction set.

Some quantity-to-quantity ratios, such as the results obtained by measuring the diagonal of a square, cannot be expressed by rational numbers. In order to solve this contradiction, irrational numbers are introduced. The so-called irrational number is an infinite cyclic decimal. The set of rational numbers and the set of irrational numbers together form the set of real numbers R. Because rational numbers can be regarded as cyclic decimals (including integers and finite decimals) and irrational numbers are infinite cyclic decimals,

Due to the needs of production and scientific development, several episodes have gradually expanded. For mathematics itself, it also solves the contradiction that some operations in the original number set can never be realized. Fractions solve the contradiction that cannot be divisible in integer sets, negative numbers solve the contradiction that cannot be reduced in positive rational numbers, and irrational numbers solve the contradiction that cannot be opened by roots. But after the number set is expanded to the real number set R, the equation like x2=- 1 still has no solution, because the square of no real number is equal to-1. Because of the need to solve equations, people introduced a new number, called imaginary unit, so complex numbers came into being.

Second, the new curriculum research:

1, imaginary unit:

(1) Its square is equal to-1, that is;

(2) Real numbers can be used for four operations, and the original laws of addition and multiplication still hold.

2. Relationship with-1: it is the square root of-1, that is, one root of equation x2=- 1, and the other root of equation x2=- 1 is-!

2. The periodicity of: 4n+ 1 = i, 4n+2 =- 1, 4n+3 =-i, 4n = 1.

3. Definition of complex number: The number in a shape is called a complex number, called the real part of a complex number, and the set formed by all complex numbers called the imaginary part of a complex number is called a complex set, which is represented by the letter c *.

4. Algebraic form of complex numbers: Complex numbers are usually represented by the letter Z, that is, the form of a+bi is called algebraic form of complex numbers.

5. Relationship between complex number and real number, imaginary number, pure imaginary number and 0: For complex number, if and only if b=0, complex number a+bi(a, b? R) is a real number a; When b? 0, the complex number z=a+bi is called 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.

6. the relationship between complex number set and other number sets: n z q r c.

7. Definition of equality of two complex numbers: If the real and imaginary parts of two complex numbers are equal, then we say that the two complex numbers are equal.

That is to say, if a, b, c, d? R, then a+bi =c+ Adi = c, B = D.

The definition of complex number equality is to find the value of complex number, which is an important basis for solving equations in complex number set. Generally speaking, two complex numbers can only be said to be equal or unequal, and they cannot be compared in size. For example, 3+5i and 4+3i are not comparable in size.

There is a proposition:? No two complex numbers can compare sizes? Right? No, if both complex numbers are real numbers, you can compare the sizes. Only when two complex numbers are not real numbers can the sizes be compared.

Illustration

Example 1 Please tell the real part and imaginary part of the complex number. Is there a pure imaginary number?

A: They are all imaginary numbers, and the real parts are 2, -3, 0,-and so on. The imaginary part is 3,,-,-; -i is a pure imaginary number.

Example 2 What are the real and imaginary parts of the complex number -2i+3. 14?

Answer: The real part is 3. 14 and the imaginary part is -2.

Error-prone: the real part is -2 and the imaginary part is 3. 14!

Example 3 What value does the real number m take? The complex number z=m+ 1+(m- 1)i is:

(1) real number? (2) imaginary number? (3) pure imaginary number?

[resolution] because m? R, so both m+ 1 and m- 1 are real numbers, and the value of m can be determined by the condition that the complex number z=a+bi is real, imaginary and pure imaginary.

Solution: (1) When m- 1=0, that is, m= 1, the complex number Z is a real number;

(2) When m- 1? 0, which is m? 1, the complex number z is imaginary;

(3) When m+ 1=0 and m- 1? 0, that is, m=- 1, and the complex number z is purely imaginary.

Example 4 (2x- 1)+i=y-(3-y)i, where x, y? R, find x and y.

Solution: According to the definition of complex number equality, the equation is obtained, so x=, y=4.

Classroom integration

1. Let the set C={ complex number}, A={ real number} and B={ pure imaginary number}. If the complete set of S=C, the following conclusion is correct ().

A. answer? B = C B A = B C A？ B= D.B？ B=C

2. If the complex number (2x2+5x+2)+(x2+x-2)i is an imaginary number, then the real number x satisfies ().

A.x=- B.x=-2 or -C.x? -2 D.x？ 1 and x? -2

3. Complex number z 1=a+|b|i, z2=c+|d|i(a, b, c, d? R), then the necessary and sufficient condition for z 1=z2 is _ _ _ _ _.

4. known m? R, complex number z =+(m2+2m-3) When is the value of I and m, (1)z? r； (2)z is an imaginary number; (3)z is a pure imaginary number; (4)z= +4i。

Induction reflex

After-class inquiry

1, let the complex number z=log2(m2-3m-3)+ilog2(3-m)(m? R), if z is pure imaginary, find the value of m 。

2. If the equation x2+(m+2i)x+(2+mi)=0 has at least one real root, try to find the value of the real number m. 。