Lesson plan.
1. Organizational moment.
2. Presentation of the material.
3. Homework.
4. Summing up the lesson.
During the classes
I. Organizational moment.
II. Presentation of the material.
Motivation.
The expansion of the set of real numbers consists of adding new numbers (imaginary) to the real numbers. The introduction of these numbers is due to the impossibility of extracting the root of a negative number in the set of real numbers.
Introduction to the concept of a complex number.
Imaginary numbers, with which we complement real numbers, are written in the form bi, Where i is an imaginary unit, and i 2 = - 1.
Based on this, we obtain the following definition of a complex number.
Definition. A complex number is an expression of the form a+bi, Where a And b- real numbers. In this case, the following conditions are met:
a) Two complex numbers a 1 + b 1 i And a 2 + b 2 i equal if and only if a 1 =a 2, b 1 =b 2.
b) The addition of complex numbers is determined by the rule:
(a 1 + b 1 i) + (a 2 + b 2 i) = (a 1 + a 2) + (b 1 + b 2) i.
c) Multiplication of complex numbers is determined by the rule:
(a 1 + b 1 i) (a 2 + b 2 i) = (a 1 a 2 - b 1 b 2) + (a 1 b 2 - a 2 b 1) i.
Algebraic form of a complex number.
Writing a complex number in the form a+bi is called the algebraic form of a complex number, where A– real part, bi is the imaginary part, and b– real number.
Complex number a+bi is considered equal to zero if its real and imaginary parts are equal to zero: a = b = 0
Complex number a+bi at b = 0 considered to be the same as a real number a: a + 0i = a.
Complex number a+bi at a = 0 is called purely imaginary and is denoted bi: 0 + bi = bi.
Two complex numbers z = a + bi And = a – bi, differing only in the sign of the imaginary part, are called conjugate.
Operations on complex numbers in algebraic form.
You can perform the following operations on complex numbers in algebraic form.
1) Addition.
Definition. Sum of complex numbers z 1 = a 1 + b 1 i And z 2 = a 2 + b 2 i is called a complex number z, the real part of which is equal to the sum of the real parts z 1 And z 2, and the imaginary part is the sum of the imaginary parts of numbers z 1 And z 2, that is z = (a 1 + a 2) + (b 1 + b 2)i.
Numbers z 1 And z 2 are called terms.
Addition of complex numbers has the following properties:
1º. Commutativity: z 1 + z 2 = z 2 + z 1.
2º. Associativity: (z 1 + z 2) + z 3 = z 1 + (z 2 + z 3).
3º. Complex number –a –bi called the opposite of a complex number z = a + bi. Complex number, opposite of complex number z, denoted -z. Sum of complex numbers z And -z equal to zero: z + (-z) = 0
Example 1: Perform addition (3 – i) + (-1 + 2i).
(3 – i) + (-1 + 2i) = (3 + (-1)) + (-1 + 2) i = 2 + 1i.
2) Subtraction.
Definition. Subtract from a complex number z 1 complex number z 2 z, What z + z 2 = z 1.
Theorem. The difference between complex numbers exists and is unique.
Example 2: Perform a subtraction (4 – 2i) - (-3 + 2i).
(4 – 2i) - (-3 + 2i) = (4 - (-3)) + (-2 - 2) i = 7 – 4i.
3) Multiplication.
Definition. Product of complex numbers z 1 =a 1 +b 1 i And z 2 =a 2 +b 2 i is called a complex number z, defined by the equality: z = (a 1 a 2 – b 1 b 2) + (a 1 b 2 + a 2 b 1)i.
Numbers z 1 And z 2 are called factors.
Multiplication of complex numbers has the following properties:
1º. Commutativity: z 1 z 2 = z 2 z 1.
2º. Associativity: (z 1 z 2)z 3 = z 1 (z 2 z 3)
3º. Distributivity of multiplication relative to addition:
(z 1 + z 2) z 3 = z 1 z 3 + z 2 z 3.
4º. z = (a + bi)(a – bi) = a 2 + b 2- real number.
In practice, multiplication of complex numbers is carried out according to the rule of multiplying a sum by a sum and separating the real and imaginary parts.
In the following example, we will consider multiplying complex numbers in two ways: by rule and by multiplying sum by sum.
Example 3: Do the multiplication (2 + 3i) (5 – 7i).
1 way. (2 + 3i) (5 – 7i) = (2× 5 – 3× (- 7)) + (2× (- 7) + 3× 5)i = = (10 + 21) + (- 14 + 15 )i = 31 + i.
Method 2. (2 + 3i) (5 – 7i) = 2× 5 + 2× (- 7i) + 3i× 5 + 3i× (- 7i) = = 10 – 14i + 15i + 21 = 31 + i.
4) Division.
Definition. Divide a complex number z 1 to a complex number z 2, means to find such a complex number z, What z · z 2 = z 1.
Theorem. The quotient of complex numbers exists and is unique if z 2 ≠ 0 + 0i.
In practice, the quotient of complex numbers is found by multiplying the numerator and denominator by the conjugate of the denominator.
Let z 1 = a 1 + b 1 i, z 2 = a 2 + b 2 i, Then
.
In the following example, we will perform division using the formula and the rule of multiplication by the conjugate number of the denominator.
Example 4. Find the quotient 
.
5) Raising to a positive whole power.
a) Powers of the imaginary unit.
Taking advantage of equality i 2 = -1, it is easy to define any positive integer power of the imaginary unit. We have:
i 3 = i 2 i = -i,
i 4 = i 2 i 2 = 1,
i 5 = i 4 i = i,
i 6 = i 4 i 2 = -1,
i 7 = i 5 i 2 = -i,
i 8 = i 6 i 2 = 1 etc.
This shows that the degree values i n, Where n– a positive integer, periodically repeated as the indicator increases by 4 .
Therefore, to raise the number i to a positive whole power, we must divide the exponent by 4 and build i to a power whose exponent is equal to the remainder of the division.
Example 5: Calculate: (i 36 + i 17) i 23.
i 36 = (i 4) 9 = 1 9 = 1,
i 17 = i 4 × 4+1 = (i 4) 4 × i = 1 · i = i.
i 23 = i 4 × 5+3 = (i 4) 5 × i 3 = 1 · i 3 = - i.
(i 36 + i 17) · i 23 = (1 + i) (- i) = - i + 1= 1 – i.
b) Raising a complex number to a positive integer power is carried out according to the rule for raising a binomial to the corresponding power, since it is a special case of multiplying identical complex factors.
Example 6: Calculate: (4 + 2i) 3
(4 + 2i) 3 = 4 3 + 3× 4 2 × 2i + 3× 4× (2i) 2 + (2i) 3 = 64 + 96i – 48 – 8i = 16 + 88i.
FEDERAL AGENCY FOR EDUCATION
STATE EDUCATIONAL INSTITUTION
HIGHER PROFESSIONAL EDUCATION
"VORONEZH STATE PEDAGOGICAL UNIVERSITY"
DEPARTMENT OF AGLEBRA AND GEOMETRY
Complex numbers
(selected tasks)
GRADUATE QUALIFICATION WORK
specialty 050201.65 mathematics
(with additional specialty 050202.65 computer science)
Completed by: 5th year student
physical and mathematical
faculty
Scientific adviser:
VORONEZH – 2008
1. Introduction……………………………………………………...…………..…
2. Complex numbers (selected problems)
2.1. Complex numbers in algebraic form….……...……….….
2.2. Geometric interpretation of complex numbers…………..…
2.3. Trigonometric form of complex numbers
2.4. Application of the theory of complex numbers to the solution of equations of the 3rd and 4th degree……………..……………………………………………………………
2.5. Complex numbers and parameters…………………………………...….
3. Conclusion……………………………………………………………………………….
4. List of references………………………….………………………......
1. Introduction
In the school mathematics curriculum, number theory is introduced using examples of sets of natural numbers, integers, rationals, irrationals, i.e. on the set of real numbers, the images of which fill the entire number line. But already in the 8th grade there is not enough supply of real numbers, solving quadratic equations with a negative discriminant. Therefore, it was necessary to replenish the stock of real numbers with the help of complex numbers, for which the square root of a negative number makes sense.
The choice of the topic “Complex numbers” as the topic of my final qualification work is that the concept of a complex number expands students’ knowledge about number systems, about solving a wide class of problems of both algebraic and geometric content, about solving algebraic equations of any degree and about solving problems with parameters.
This thesis examines the solution to 82 problems.
The first part of the main section “Complex numbers” provides solutions to problems with complex numbers in algebraic form, defines the operations of addition, subtraction, multiplication, division, the conjugation operation for complex numbers in algebraic form, the power of an imaginary unit, the modulus of a complex number, and also sets out the rule extracting the square root of a complex number.
In the second part, problems on the geometric interpretation of complex numbers in the form of points or vectors of the complex plane are solved.
The third part examines operations on complex numbers in trigonometric form. The formulas used are: Moivre and extracting the root of a complex number.
The fourth part is devoted to solving equations of the 3rd and 4th degrees.
When solving problems in the last part, “Complex numbers and parameters,” the information given in the previous parts is used and consolidated. A series of problems in the chapter are devoted to determining families of lines in the complex plane defined by equations (inequalities) with a parameter. In part of the exercises you need to solve equations with a parameter (over field C). There are tasks where a complex variable simultaneously satisfies a number of conditions. A special feature of solving problems in this section is the reduction of many of them to the solution of equations (inequalities, systems) of the second degree, irrational, trigonometric with a parameter.
A feature of the presentation of the material in each part is the initial introduction of theoretical foundations, and subsequently their practical application in solving problems.
At the end of the thesis there is a list of references used. Most of them present theoretical material in sufficient detail and in an accessible manner, discuss solutions to some problems, and give practical tasks for independent solution. I would like to pay special attention to such sources as:
1. Gordienko N.A., Belyaeva E.S., Firstov V.E., Serebryakova I.V. Complex numbers and their applications: Textbook. . The material of the textbook is presented in the form of lectures and practical exercises.
2. Shklyarsky D.O., Chentsov N.N., Yaglom I.M. Selected problems and theorems of elementary mathematics. Arithmetic and algebra. The book contains 320 problems related to algebra, arithmetic and number theory. These tasks differ significantly in nature from standard school tasks.
2. Complex numbers (selected problems)
2.1. Complex numbers in algebraic form
The solution of many problems in mathematics and physics comes down to solving algebraic equations, i.e. equations of the form
,where a0, a1, …, an are real numbers. Therefore, the study of algebraic equations is one of the most important issues in mathematics. For example, a quadratic equation with a negative discriminant has no real roots. The simplest such equation is the equation
.In order for this equation to have a solution, it is necessary to expand the set of real numbers by adding to it the root of the equation
.Let us denote this root by
. Thus, by definition, or,hence,
. called the imaginary unit. With its help and with the help of a pair of real numbers, an expression of the form is compiled.The resulting expression was called complex numbers because they contained both real and imaginary parts.
So, complex numbers are expressions of the form
, and are real numbers, and is a certain symbol that satisfies the condition . The number is called the real part of a complex number, and the number is its imaginary part. The symbols , are used to denote them.Complex numbers of the form
are real numbers and, therefore, the set of complex numbers contains the set of real numbers.Complex numbers of the form
are called purely imaginary. Two complex numbers of the form and are said to be equal if their real and imaginary parts are equal, i.e. if equalities , .Algebraic notation of complex numbers allows operations on them according to the usual rules of algebra.
The sum of two complex numbers
and is called a complex number of the form .Product of two complex numbers
Let us recall the necessary information about complex numbers.
Complex number is an expression of the form a + bi, Where a, b are real numbers, and i- so-called imaginary unit, a symbol whose square is equal to –1, that is i 2 = –1. Number a called real part, and the number b - imaginary part complex number z = a + bi. If b= 0, then instead a + 0i they simply write a. It can be seen that real numbers are a special case of complex numbers.
Arithmetic operations on complex numbers are the same as on real numbers: they can be added, subtracted, multiplied and divided by each other. Addition and subtraction occur according to the rule ( a + bi) ± ( c + di) = (a ± c) + (b ± d)i, and multiplication follows the rule ( a + bi) · ( c + di) = (ac – bd) + (ad + bc)i(here it is used that i 2 = –1). Number = a – bi called complex conjugate To z = a + bi. Equality z · = a 2 + b 2 allows you to understand how to divide one complex number by another (non-zero) complex number:
(For example, 
.)
Complex numbers have a convenient and visual geometric representation: number z = a + bi can be represented by a vector with coordinates ( a; b) on the Cartesian plane (or, which is almost the same thing, a point - the end of a vector with these coordinates). In this case, the sum of two complex numbers is depicted as the sum of the corresponding vectors (which can be found using the parallelogram rule). According to the Pythagorean theorem, the length of the vector with coordinates ( a; b) is equal to . This quantity is called module complex number z = a + bi and is denoted by | z|. The angle that this vector makes with the positive direction of the x-axis (counted counterclockwise) is called argument complex number z and is denoted by Arg z. The argument is not uniquely defined, but only up to the addition of a value that is a multiple of 2 π
radians (or 360°, if counted in degrees) - after all, it is clear that turning by such an angle around the origin will not change the vector. But if the vector of length r forms an angle φ
with the positive direction of the x-axis, then its coordinates are equal to ( r cos φ
; r sin φ
). From here it turns out trigonometric notation complex number: z = |z| · (cos(Arg z) + i sin(Arg z)). It is often convenient to write complex numbers in this form, because it greatly simplifies the calculations. Multiplying complex numbers in trigonometric form is very simple: z 1 · z 2 = |z 1 | · | z 2 | · (cos(Arg z 1 + Arg z 2) + i sin(Arg z 1 + Arg z 2)) (when multiplying two complex numbers, their modules are multiplied and their arguments are added). From here follow Moivre's formulas: z n = |z|n· (cos( n· (Arg z)) + i sin( n· (Arg z))). Using these formulas, it is easy to learn how to extract roots of any degree from complex numbers. nth root of z- this is a complex number w, What w n = z. It's clear that 
, And where k can take any value from the set (0, 1, ..., n- 1). This means that there is always exactly n roots n th degree of a complex number (on the plane they are located at the vertices of the regular n-gon).
Complex numbers
Imaginary And complex numbers. Abscissa and ordinate
complex number. Conjugate complex numbers.
Operations with complex numbers. Geometric
representation of complex numbers. Complex plane.
Modulus and argument of a complex number. Trigonometric
complex number form. Operations with complex
numbers in trigonometric form. Moivre's formula.
Basic information about imaginary And complex numbers are given in the section “Imaginary and complex numbers”. The need for these numbers of a new type arose when solving quadratic equations for the case
D< 0 (здесь D– discriminant of a quadratic equation). For a long time, these numbers did not find physical application, which is why they were called “imaginary” numbers. However, now they are very widely used in various fields of physics.and technology: electrical engineering, hydro- and aerodynamics, elasticity theory, etc.
Complex numbers are written in the form:a+bi. Here a And b – real numbers , A i – imaginary unit, i.e. e. i 2 = –1. Number a called abscissa, a b – ordinatecomplex numbera + bi.Two complex numbersa+bi And a–bi are called conjugate complex numbers.
Main agreements:
1. Real number
Acan also be written in the formcomplex number:a + 0 i or a – 0 i. For example, records 5 + 0i and 5 – 0 imean the same number 5 .2. Complex number 0 + bicalled purely imaginary number. Recordbimeans the same as 0 + bi.
3. Two complex numbersa+bi Andc + diare considered equal ifa = c And b = d. Otherwise complex numbers are not equal.
Addition. Sum of complex numbersa+bi And c + diis called a complex number (a+c ) + (b+d ) i.Thus, when adding complex numbers, their abscissas and ordinates are added separately.
This definition corresponds to the rules for operations with ordinary polynomials.
Subtraction. The difference of two complex numbersa+bi(diminished) and c + di(subtrahend) is called a complex number (a–c ) + (b–d ) i.
Thus, When subtracting two complex numbers, their abscissas and ordinates are subtracted separately.
Multiplication. Product of complex numbersa+bi And c + di is called a complex number:
(ac–bd ) + (ad+bc ) i.This definition follows from two requirements:
1) numbers a+bi And c + dimust be multiplied like algebraic binomials,
2) number ihas the main property:i 2 = – 1.
EXAMPLE ( a+ bi )(a–bi) = a 2 +b 2 . Hence, work
two conjugate complex numbers is equal to the real
a positive number.
Division. Divide a complex numbera+bi (divisible) by anotherc + di(divider) - means to find the third numbere + f i(chat), which when multiplied by a divisorc + di, results in the dividenda + bi.
If the divisor is not zero, division is always possible.
EXAMPLE Find (8 +i ) : (2 – 3 i) .
Solution. Let's rewrite this ratio as a fraction:
Multiplying its numerator and denominator by 2 + 3i
AND Having performed all the transformations, we get:
Geometric representation of complex numbers. Real numbers are represented by points on the number line:
Here is the point Ameans the number –3, dotB– number 2, and O- zero. In contrast, complex numbers are represented by points on the coordinate plane. For this purpose, we choose rectangular (Cartesian) coordinates with the same scales on both axes. Then the complex numbera+bi will be represented by a dot P with abscissa a and ordinate b (see picture). This coordinate system is called complex plane .
Module complex number is the length of the vectorOP, representing a complex number on the coordinate ( comprehensive) plane. Modulus of a complex numbera+bi denoted | a+bi| or letter r