Scientific and practical conference

"First Steps in Science"

Section" Maths"

Completed: 9th grade student MBOU

"Mordovian-Paevskaya secondary school"

Erochkin Ivan

Supervisor: mathematic teacher

Kadyshkina N.V.

Insar 2014

TABLE OF CONTENTS

Introduction…………………………………………………………………

The history of the discovery of complex numbers ………………………… 4

2.1. Sayings of great scientists about complex numbers…. 4

2.2 On the emergence of complex numbers………………………………4

Main part

Definition of complex numbers…………………………………. eight

2.1. Algebraic form of a complex number………………8

2.2. Operations on complex numbers…………………… 9

3. Solving equations with a complex variable………………… 12

4. The concept of the complex plane……………………………….. 14

5. Geometric form of a complex number…………………….. 15

6. Trigonometric form of number………………………………….. 17

7. Raising to the power of a complex number………………………. nineteen

The exponential form of the number……………………………………… 20

Where are complex numbers used? .............................................. 21

Conclusion. Conclusions……………………………………………… 23

References…………………………………………………… 24

Test on the topic “Complex numbers”………………………………. 25

Introduction In ancient times, having learned to count, people learned the measure of quantity - the number. NUMBER - one of the basic concepts of mathematics, originated in ancient times and gradually expanded and generalized. Attractive with natural beauty, filled with inner harmony, accessible, but still incomprehensible, hiding many secrets behind the seeming simplicity ... In our lives, each of us stacks with numbers. The course of the school curriculum, and further life, is hard to imagine without them.

Natural, whole, rational, irrational, real. They fascinate me more and more every year. Last year, I did research on the mysterious number pi. I was interested in complex numbers. I first heard about them in the 8th grade, solving quadratic equations. In the 9th grade, I had serious problems solving cubic equations that must have three roots, since after decomposing a polynomial into linear factors, it becomes necessary to solve a quadratic equation. And suddenly it turns out that the discriminant is negative, that is, the quadratic equation has no roots, because when finding the roots of a quadratic equation, I need to extract the arithmetic square root from a negative number. And this means that the cubic equation instead of three roots has only one root. This is how I got the contradiction. And I decided to look into it. Such an operation is impossible on the set of real numbers, but not impossible at all. It turned out that the roots of the equation I was solving belong to the set of complex numbers, which contains a number whose square is -1.My interest increased even more when I learned a lot about complex numbers.

Objective: To study complex numbers as a branch of mathematics and their role in many branches of mathematics.

Research objectives:

1. Analyze the literature on this issue;

2. Systematize information about numbers;

3. Expand number sets from natural to complex as a way

building a new mathematical apparatus.

4. Improve the technique of algebraic transformations.

5. Assess the importance and role of complex numbers in mathematics, in increasing interest in the study of complex numbers by 9th grade students, in developing their creative and research abilities.

Problem: the absence in the programs for the course of algebra and the beginning of analysis for general educational institutions of a section that studies complex numbers.

Working hypothesis: It is assumed that the acquaintance and study of complex numbers by students will allow them to deepen their knowledge in many sections of mathematics, equip them with an additional tool for solving various problems.

Subject of study: complex numbers.

Object of study: forms for specifying a complex number and actions on them.

Research methods:

1. Study and analysis of literary sources.

2. Solution of practical problems

3. Develop a test.

4. Survey.

5. Analysis of the work done.

Relevance of the topic.

I think my subjectrelevant , since although in our time there is quite a lot of scientific and educational literature, but not in all publications the material is presented clearly, understandably and accessible to us students. My interest increased even more when I learned a lot about complex numbers. Here is the result of my work on this topic.

Main part.

History of the discovery of complex numbers

Imaginary numbers are a beautiful and wonderful refuge of the divine spirit. almost an amphibian with nothingness. G. Leibniz

“In addition to and even against the will of this or that mathematician, imaginary numbers appear again and again in the calculations, and only gradually, as the benefits of their use are discovered, do they become more and more widespread ”F. Klein.

After all, no one doubts the accuracy of the results obtained in calculations with imaginary quantities, although they are only algebraic forms and hieroglyphs of ridiculous quantities.

L. Carnot

The process of expanding the concept of number from natural to real was connected both with the needs of practice and with the needs of mathematics itself. Ancient Greek scientists considered only natural numbers to be “real”, but in practical calculations for two millennia BC. fractions were already used in ancient Babylon and ancient Egypt. The next important stage in the development of the concept of number was the appearance of negative values. They were introduced by Chinese scientists two centuries BC. e., and the ancient Greek mathematician Diophantus in III century AD e. already knew how to perform actions on negativesolid numbers.

In mathematics, they are called the set of real numbers.

All real numbers are located on the number line:

The company of real numbers is very colorful - here are whole numbers, and fractions, irrational numbers. In this case, each numeric point necessarily corresponds to some real number.

V XIII century began to extract square rootsfrom positive numbers and found that with numbers negativethis operation is not possible. But inXVI century in connection with the studycubic equations of mathematics encountered a problem:in connection with the study of cubic equations, it turned out to be necessary to extract square roots from negative numbers.

Atalignment mustnamebethree roots. When solving it, oftena negative number appeared under the square root sign. It turned out that the path to these roots leads through the impossible operation of extracting the square root of a negative number.

To explain the resulting paradox, the Italian algebraist Girolamo Cardano in 1545 proposed introducing numbers of a new nature. He showed that the system of equations x + y = 10, xy = 40 which has no solutions in the set of real numbers, always has a solution x = 5 ±
, y = 5 ±
, we only need to agree to act on such expressions according to the rules of ordinary algebra and assume that
∙
= - a. Cardano called such quantities "pure negative" and even "sophistically negative",but he considered them completely useless and strove not to use them. However, already in 1572, his compatriot R. Bombelli published a book in which the first rules for arithmetic operations on such numbers were established, up to extraction fromthem cube roots.

The name "imaginary numbers" was introduced in 1637

French mathematician and philosopher R. Descartes.

And in 1777 one of the greatest algebraists XVIII century - L. Euler - suggested using the first letter of the French wordimaginaire (think my) to denote a numberi =
.

This symbol came into general use thanks to K. Gauss.The term "complex numbers ” was also introduced by Gauss in 1831. The word complex (from the Latincomplexus ) means connection, combination, set of concepts, objects, phenomena, etc., O forming a single whole.

During the XVII century continued discussion of the arithmetic nature of imaginary numbers, the possibility of giving them a geometric justification.

The technique of operations on complex numbers gradually developed. On the edge XVII - XVIII centuries, a general theory of roots was builtn th degree, first from negative, and subsequently from any complex numbers.

At the end of XVIII century, the French mathematician J. Lagrange was able to say that imaginary quantities no longer complicate mathematical analysis. With the help of complex numbers, they learned to express solutions of linear differential equations with a constant coefficient. Such equations are encountered, for example, in the theory of oscillations of a material point in a resisting medium.

J. Bernoulli used complex numbers to calculate integrals. Although during XVIII century, with the help of complex numbers, many issues were solved, including applied problems related to cartography, hydrodynamics, etc., but there was still no rigorous rationale for the theory of these numbers. Therefore, the French scientist P. Laplace believed that the results obtained with the help of imaginary numbers are only guidance, acquiring the character of real truths only after confirmation by direct evidence. In the end XVIII - early XIX centuries, a geometric interpretation of complex numbers was obtained. The Dane G. Wessel, the Frenchman J. Argan and the German K. Gauss independently proposed to represent the complex number z \u003d a + bi point M (a, b ) on the coordinate plane. Later it turned out that it was even more convenient to represent the number not by the point M itself, but by the vector OM going to this point from the origin. With this interpretation, addition and subtraction of complex numbers correspond to the same operations on vectors.

Geometric interpretations of complex numbers made it possible to define many concepts related to the functions of a complex variable, and expanded the scope of their application. It became clear that complex numbers are useful in many issues where they deal with quantities that are represented by vectors on a plane: in the study of fluid flow, problems in the theory of elasticity, in theoretical electrical engineering.

A great contribution to the development of the theory of functions of a complex variable was made by Russian and Soviet scientists: R.I. Muskhelishvili was engaged in its applications to the theory of elasticity, M.V. Keldysh and M.A. Lavrentiev - to aerodynamics and hydrodynamics, N.N. Bogolyubov and V.S. Vladimirov - to the problems of quantum field theory.

Definition of complex numbers

3.1 Algebraic form of a complex number

complex number z called expression z = a + b i, where a and b are real numbers,i 2 = -1,

a = Re z – real part z (real) (Re, from French ré ele - "real", "valid");

b = Im z imaginary part z (Im, from French imaginaire - “imaginary”) .

b – the coefficient of the imaginary part of a complex number.

Writing a complex number z as a + ib is called the algebraic form of the complex number.

If a 0, in 0, that number z- imaginary ( z = 37 - 6 i ).

E if a = 0 , v 0, that number z is a pure imaginary number ( z = 22 i) .

If a 0, at =0, z is a real number ( z = -5).

Powers of i:

I 1 = i
i 4n+1 = i;

i 2 = - 1
i 4n+2 = - 1;

i 3 = i 2 i
i 4n+3 = - i

i 4 = (i 2 ) 2 = 1
i 4 n = 1.

It follows from the formulas that addition and multiplication can be performed according to the rules of operations with polynomials, assuming i 2 = -1. The operations of addition and multiplication of complex numbers have the properties of real numbers. Basic properties:

Transfer property:

Z 1 + Z 2 \u003d Z 2 + Z 1, Z 1 Z 2 \u003d Z 2 Z 1

Associative property:

(Z 1 + Z 2) + Z 3 \u003d Z 1 + (Z 2 + Z 3), (Z 1 Z 2) Z 3 \u003d Z 1 (Z 2 Z 3)

Distribution property:

Z 1 (Z 2 + Z 3) \u003d Z 1 Z 2 + Z 1 Z 3

the sum of two opposite numbers is 0 (z + (- z ) = 0)

A complex number is equal to zero if the real and imaginary parts are equal to zero, respectively.

3.2 Operations on complex numbers.

Over complex numbers written in algebraic form, it is possible to carry out all arithmetic operations as over ordinary binomials, taking into account only that i 2 = -1.

Addition and subtraction of complex numbers.

Sum of complex numbers z 1 \u003d a 1 + b 1 i and z 2 \u003d a 2 - b 2 i is equal to:
z 1 + z 2 \u003d (a 1 + a 2) + (b 1 + b 2) i

Example 1

Add two complex numbersz 1 = 1 +3 i, z 2 =4-5 i

To add two complex numbers, add their real and imaginary parts:

z 1 + z 2 \u003d 1 + 3i + 4 -5i \u003d 5 -2i

Difference of complex z 1 = a 1 + b 1 i and z 2 = a 2 – b 2 i numbers equal on the:

z 1 - z 2 \u003d (a 1 - a 2) + (b 1 - b 2) i

Example 2

Find differences of complex numbersz 1 = -2 + iandz 2 = 4 i -2

The action is similar to addition, the only feature is that the subtrahend must be taken in brackets, and then, as a standard, open these brackets with a sign change:

z 1 - z 2 \u003d (-2 + i) - (4i - 2) \u003d -2 + I - 4i +2 \u003d - 3i

Multiplication of complex numbers

Product of complex numbers z 1 \u003d a 1 + b 1 i and z 2 \u003d a 2 - b 2 i is equal to:

z 1 z 2 \u003d (a 1 a 2 - b 1 b 2) + (a 2 b 1 + b 2 a 1) i

Example 3 Find the product of complex numbers

z 1 \u003d 1 - i, z 2 \u003d 3 + 6i

z 1 z 2 \u003d (1 -i) (3 + 6i) \u003d 1 3 -i 3 + 1 6i - i 6i \u003d 3- 3i + 6i +6 \u003d 9 + 3i

Division of complex numbers

Quotient of complex numbers z 1 = a 1 + b 1 · i and z 2 = a 2 – b 2 · i equals:

Example 4. Let z 1 \u003d 13 + i, z 2 \u003d 7 - 6 i

To find the quotient, first multiply the numerator and denominator of the fraction by the conjugate of the denominator, and then perform the rest of the operations.

Extraction of roots from complex numbers.

Can't extract the root? If we are talking about real numbers, then it is really impossible. In complex numbers, you can extract the root - you can! More precisely, two root:

Are the found roots really the solution of the equation? Let's check:

These roots are also called conjugate complex roots.

When extracting square roots from negative numbers, we get two conjugate complex roots.

For example, , , , ,

Solving equations with a complex variable

First, I considered the simplest quadratic equation z 2 = a , where a - a given number, z is unknown. On the set of real numbers, this equation is:

1) has one root z = 0 if a = 0;

2) has two real roots z 1,2 = ±
if a > 0;

3) has no real roots if a< 0;

4) on the set of complex numbers, this equation always has a root.

In general, the equation z 2 = a , where a < 0 имеет два комплексных корня: z 1,2 =±
i .

Using equality i 2 = -1, the square roots of negative numbers are usually written as follows:
= i ,
= i
= 2 i ,
= i
.

So,
defined for any real number a (positive, negative and zero). So any quadratic equation

az 2 + bz + c = 0, where a , b , s are real numbers, a ≠ 0, has roots. These roots are found according to the well-known formula:

z 1, 2 =
.

It is also true that any equation of degreen has exactly n roots, while among them can be the same and complex.

It is impossible not to consider one of the most beautiful formulas of mathematics - the Cardano formula for calculating the roots of a cubic equation of the form x 3 + px + q = 0:

Example 5 Solve a quadratic equation

Discriminant:

D<0, и в действительных числах уравнение решения не имеет. Но корень можно извлечь в комплексных числах!

There are two roots:

are conjugate complex roots

So the equation has two conjugate complex roots: ,

And in general, any equation with a polynomial of "nth" degree has exactly roots, some of which may be complex.

The concept of the complex plane.

If any real number can be geometrically represented as a point on a number line, then a complex number is represented by a point on the plane, the coordinates of which will be, respectively, the real and imaginary parts of the complex number Letter R it is customary to denote the set of real numbers.A bunch ofcomplex numbers usually denoted by the letter C. In this case, the horizontal axis will be the real numerical axis, and the vertical axis will be the imaginary axis.

Thus, real numbers are located on the x-axis, and on the o-axis Y are purely imaginary:

The rules for designing a drawing are almost the same as for a drawing in a Cartesian coordinate system. Along the axes, you need to set the dimension, note: zero; unit along the real axis; imaginary unit along the imaginary axis.

Example 6. Construct the following complex numbers on the complex plane:

The set of real numbersis a subset of the set of complex numbers.

6. The geometric form of a complex number.

WITH The word “complex” in Latin means “composite”, “complex”. Despite the fact that complex numbers are no more difficult to work with than real numbers, until the early nineteenth century, complex numbers were viewed as a very complex, obscure, almost mystical object. With tenacity worthy of a better use, a long struggle was waged between supporters and opponents of "imaginary" numbers. The main objection of the opponents was as follows: an expression of the form a+ib meaningless because i is not a real number, and therefore is not a number at all; That's why i cannot be multiplied by a real number.

To put the theory of complex numbers on a solid foundation, its explicit construction was necessary, preferably geometric. The desire to have a geometric realization of the set of complex numbers is not accidental, if we recall that the set of real numbers is also inseparable for us from the “real line” with a fixed point representing 0 on it, and with a fixed scale determined by the position of the number 1.

For the first time, the representation of geometric operations on complex numbers was given by the Danish surveyor K. Wessel in 1799 and independently by the French mathematician J. Argan in 1806. However, it received general recognition only in the thirties of the eighteenth century after the work of the German mathematician F. Gauss and the English mathematician W. Hamilton. The idea of the geometric interpretation of complex numbers is that they are represented not by points of a straight line, like real numbers, but by points of a plane.

Complex numberz = a + b i is depicted on a plane with Cartesian rectangular coordinates by a point having coordinates (a;b). This

point is denoted by the same letterz . Real numbers are represented by points of the abscissa axis, and purely imaginary numbers by points of the ordinate axis.

A complex number is also represented as a vector on the complex plane with origin at the point O and end at point M.

The sum of complex numbers is constructed according to the usual rule of vector addition, that is, according to the parallelogram rule

The difference of complex numbers is built according to the vector subtraction rule:

7.Trigonometric form of a complex number.

Arbitrary complex number z = a + bi represented as a radius vector
on the complex plane. Let N – point projection M to the real axis. In a right triangle OMN leg lengths ON and OM equal respectively a and b , and the length of the hypotenuse OM is
. From trigonometry it is known that the ratio of the length of the leg to the length of the hypotenuse is equal to the cosine of the adjacent angle and the sine of the opposite. Hence,

a = Rez = | z | ∙ cos φ ,

b = Im z = | z | ∙ sin φ ,

where φ –
- the main argument (phase, amplitude) of the complex number z , - < φ < (injectionφ between positive semiaxis of the real axis Rez and the radius vector drawn from the origin to the corresponding point). Then the complex number can be represented as:

This form of writing is called trigonometric form of a complex number.

Example 7:Solution:
Let's represent the number in trigonometric form. Find its modulus and argument. . Since (case 1), then . In this way: is a number in trigonometric form.

Product and quotient of complex numbers in trigonometric form

All algebraic operations with complex numbers given in trigonometric form, are performed according to the same rules as with complex numbers given in algebraic form. Adding and subtracting complex numbers is easier and more convenient when they are given in algebraic form, and multiplying and dividing in trigonometric form. There are three theorems.

Theorem 1. When multiplying any finite number of complex numbers, their moduli are multiplied and the arguments are added.

Theorem 2. When dividing complex numbers, their moduli are divided and the arguments are subtracted.

Theorem 3. Let z is complex, and n - natural number. In the set of complex numbers, the expression
at z =0 has a single value equal to zero, and when z 0-n different values. If z = r ( cos +i sin ), then these values are found by the formula

=
(cos
+i sin
), \u003d 0.1, ..., n -1.

Example 8. Find a product: ,

8. Raising complex numbers to a power

Squaring a complex number

For a complex number, it is easy to derive your own abbreviated multiplication formula:
. A similar formula can be derived for the square of the difference, as well as for the cube of the sum and the cube of the difference. What if a complex number needs to be raised to, say, the 5th, 10th, or 100th power? It is clear that in algebraic form it is practically impossible to perform such an action, indeed, how to solve an example like ?

And here the trigonometric form of a complex number comes to the rescue and the so-called De Moivre's formula.

(Abraham de Moivre (1667 - 1754) - English mathematician).

From the operation of multiplication of complex numbers it follows that

In general, we get:

where n – a positive integer.

Example 7. Given a complex number , find .

First you need to represent the given number in trigonometric form.

Then, according to De Moivre's formula:

9. The exponential form of a complex number

=8 + 6 i

10. Where are complex numbers used?

Over the past two hundred years, complex numbers have found numerous, and sometimes completely unexpected applications. So, for example, with the help of complex numbers, Gauss found an answer to a purely geometric question: for which natural n can a regular n-gon be constructed with a compass and a ruler? From the school geometry course, it is known how to build some regular polygons with a compass and a ruler: a regular triangle, a square, a regular 6-gon (its side is equal to the radius of the circle circumscribed around it). More difficult is the construction of regular 5-gon and 15-gon. Despite the great efforts of many remarkable ancient Greek geometers and other scientists, no one managed to build either a regular heptagon or a regular 9-holed. It was also not possible to construct a regular p-gon for any prime number p, except for p = 3 and p = 5. For more than two thousand years, no one could advance in solving this problem. In 1796, Carl Friedrich Gauss, a 19-year-old mathematics student at the University of Göttingen, first proved the possibility of constructing a regular 17-gon using a compass and straightedge. It was one of the most amazing discoveries in the history of mathematics. Over the next few years, Gauss completely solved the problem of constructing regular n-gons. Gauss proved that a regular N-gon with an odd number of sides (vertices) can be constructed using a compass and straightedge if and only if N is a Fermat prime or a product of several different Fermat primes. (Fermat numbers are numbers of the form F n \u003d + 1 For n \u003d 0, 1, 2, 3, 4 these numbers are prime, for n \u003d 5 the number F 5 will be composite. From this result it followed that the construction of a regular polygon is impossible for N = 7, 9, 11, 13. It is easy to see that the problem of constructing a regular n-gon is equivalent to the problem of dividing a circle of radius R = 1 into n equal parts. degrees from unity.

The theory of functions of a complex variable is widely used in solving important practical problems of cartography, electrical engineering, thermal conductivity, etc. In many issues, where we are talking, for example, about the electric potential at points in the space surrounding a charged capacitor, or about the temperature inside a heated body, about speeds particles of a liquid or gas in a flow moving in a certain channel and flowing around some obstacles, etc., one must be able to find the potential, temperature, velocities, etc. Problems of this kind can be solved without much difficulty in the case when the bodies encountered in them have a simple shape (for example, in the form of flat plates or circular cylinders).

Russian and Soviet scientist H. E. Zhukovsky (1847–1921) successfully applied

the theory of functions of a complex variable to the solution of important applied problems.

So, using the methods of this theory, he proved the main theorem about the lift force of an aircraft wing. V. I. Lenin called H. E. Zhukovsky "the father of Russian aviation." In one of his speeches, H. E. Zhukovsky said: “... a person does not have wings and, in relation to the weight of his body to the weight of muscles, he is 72 times weaker than a bird; ... it is almost 800 times heavier than air, while a bird is 200 times heavier than air. But, I think that he will fly, relying not on the strength of his muscles, but on the strength of his mind. With the help of the theory of functions of a complex variable, H.E. Zhukovsky solved problems related to the issues of water seepage through dams.

Complex numbers are needed to complete the tasks of other sections of higher mathematics, in addition, they are used in quite material engineering calculations in practice.

11. Conclusion

In general, I believe that the purpose and tasks of her work have been fulfilled. I have mastered the topic myself. In the course of the study, I studied a lot of literature on this topic. In the course of reading various books, I noted for myself the most interesting, simple and beautiful facts on this topic, while at the same time trying to present them in my own light, in the way that I consider the most rational.

The advantages of my work include brevity and simplicity of presentation, unification of knowledge about complex numbers together, accessibility.

I find my work useful and relevant for those students who want to learn more about the school curriculum.

During the course of the study, I conducted several sessions in my class. But since there are only 2 students in our class besides me, it was not possible to trace the improvement in the quality of knowledge, since they are doing well. But I am glad that everyone wished to continue studying this topic in the 10th grade.

My Findings:

1. Various literary sources have been studied, material has been selected that gives the most complete picture of complex numbers, the history of their discovery, their role and significance in various branches of mathematics. The arithmetic operations performed on these numbers are defined and considered, examples are selected and solved using complex numbers.

2. The significance and role of complex numbers in solving a number of mathematical problems is estimated.

3. If at the beginning of the school year the level of awareness and knowledge among 9th grade students about complex numbers can be assessed as low, then by the end of the school year there was an increase in interest in studying mathematics, broadening their horizons, and successfully solving many problems of an increased level of complexity.

12. References

1. A.G. Mordkovich. Algebra and the beginnings of analysis. 10 cells Moscow: Mnemosyne, 2006.

2. M. Ya. Vygodsky; Handbook of elementary mathematics. M.: State publishing house of physical and mathematical literature, 1960.

3. N.Ya. Vilenkin et al. Algebra and mathematical analysis. 11 cells Moscow: Mnemosyne, 2004.

4. A.G. Mordkovich. Algebra and the beginnings of analysis. 10 cells Moscow: Mnemosyne, 2006.

5 . History of mathematics at school, edited by G. I. Glazer. - Moscow-1983.

6. Selected questions of mathematics, edited by IN Antipov. - Moscow-1979.

7. Behind the pages of a mathematics textbook edited by N. Ya. Vilenkin. - Moscow-1996.

8. N.B. Alfutov. Algebra and number theory. M.: MTsNMO, 2005.

Test on the topic "Complex numbers"

How many notations does a complex number have?

a) 1 b) 2 c) 3 d) 4

What is a number i?

a) a number whose square is 1

b) a number whose square is - 1

c) a number whose square root is - 1

d) a number whose square root is 1

De Moivre's formula can be applied if the complex number is written:

The Euler formula can be applied if the complex number is written:

a) in an indicative form b) in a visual form

c) trigonometric form d) algebraic form

How is a complex number represented on the number plane?

a) as a segment b) as a point or radius vector

c) a flat geometric figure c) in the form of a circle

Choose from the given numbers purely imaginary:

a) z =3 +6 i b) z 2 =6 i in) z 2 =31 g) z 2 =0

Calculate the sum of numbers z 1 =7 +2i and z 2 =3 +7 i

a ) z =10 +9i b) z =4-5i c) z =10 -5i d) z =4 +5i

8. Represent the complex number z \u003d 3 + 4i in trigonometric form

a) is the radius vector b) z =5(0.6 +0.8i )

in) z =3 -4i d) is a point on the coordinate plane

9. Which set includes the numbers 5; 3; -6i ;2.7; 2i?

a) real numbers b) rational numbers

c) complex numbers d) irrational numbers

10. Who introduced the name "imaginary numbers"?

a) Descartes b) Argan

c) Euler d) Cardano

HISTORY REFERENCE

Complex numbers were introduced into mathematics in order to make it possible to take the square root of any real number. This, however, is not sufficient reason to introduce new numbers into mathematics. It turned out that if you perform calculations according to the usual rules on expressions in which the square root of a negative number occurs, then you can come to a result that no longer contains the square root of a negative number. In the XVI century. Cardano found a formula for solving a cubic equation. It turned out that when a cubic equation has three real roots, the square root of a negative number occurs in the Cardano formula. Therefore, the square roots of negative numbers began to be used in mathematics and called them imaginary numbers - thereby, as it were, they acquired the right to an illegal existence. Gauss gave full civil rights to imaginary numbers, who called them complex numbers, gave a geometric interpretation, and proved the fundamental theorem of algebra, which states that every polynomial has at least one real root.

1. THE CONCEPT OF A COMPLEX NUMBER

The solution of many problems in mathematics and physics is reduced to solving algebraic equations. Therefore, the study of algebraic equations is one of the most important questions in mathematics. The desire to make equations solvable is one of the main reasons for expanding the concept of number.

So, for the solvability of equations of the form X+A=B, positive numbers are not enough. For example, the equation X+5=2 has no positive roots. Therefore, you have to enter negative numbers and zero.

On the set of rational numbers, algebraic equations of the first degree are solvable, i.e. equations of the form A · X+B=0 (A0). However, algebraic equations of degree higher than one may not have rational roots. For example, such are the equations X 2 =2, X 3 =5. The need to solve such equations was one of the reasons for the introduction of irrational numbers. Rational and irrational numbers form the set of real numbers.

However, real numbers are not enough to solve any algebraic equation. For example, a quadratic equation with real coefficients and a negative discriminant has no real roots. The simplest of them is the equation X 2 +1=0. Therefore, it is necessary to expand the set of real numbers by adding new numbers to it. These new numbers, together with the real numbers, form a set, which is called the set complex numbers.

Let us first find out what form the complex numbers should have. We assume that on the set of complex numbers the equation X 2 +1=0 has a root. We denote this root by the letter i In this way, i is a complex number such that i 2 = –1.

As for real numbers, one must introduce the operations of addition and multiplication of complex numbers so that their sum and product would be complex numbers. Then, in particular, for any real numbers A and B the expression A + B i can be thought of as a general representation of a complex number. The name “complex” comes from the word “composite”: by the form of the expression A + B i .

Complex numbers are called expressions of the form A+B i , where A and B are real numbers, and i is some character such that i 2 = –1, and denoted by the letter Z.

The number A is called the real part of the complex number A+B i, and the number B is its imaginary part. Number i is called the imaginary unit.

For example, the real part of the complex number 2+3 i is 2 and the imaginary is 3.

For a rigorous definition of a complex number, it is necessary to introduce the concept of equality for these numbers.

Two complex numbers A+B i and C+D i called equal if and only if A=C and B=D, i.e. when their real and imaginary parts are equal.

2. GEOMETRIC INTERPRETATION OF A COMPLEX NUMBER

Real numbers are geometrically represented by points on the number line. Complex number A+B i can be viewed as a pair of real numbers (A;B). Therefore, it is natural to represent a complex number as points in the plane. In a rectangular coordinate system, the complex number Z=A+B· i is represented by a plane point with coordinates (A;B), and this point is denoted by the same letter Z (Figure 1). Obviously, the correspondence obtained in this case is one-to-one. It makes it possible to interpret complex numbers as points on the plane on which the coordinate system is chosen. This coordinate plane is called complex plane . The abscissa is called real axis , because on it there are points corresponding to real numbers. The y-axis is called imaginary axis – it contains points corresponding to imaginary complex numbers.

No less important and convenient is the interpretation of the complex number A+B i as a vector, i.e. vectors with origin at a point

O(0;0) and ending at the point M(A;B) (Figure 2).

The correspondence established between the set of complex numbers, on the one hand, and the sets of points or vectors of the plane, on the other, allows complex numbers to be points or vectors.

3.MODULE OF A COMPLEX NUMBER

Let a complex number Z=A+B· i . Conjugated from Z is called a complex number A – B i , which is denoted by , i.e.

A-B i .

Note that = A + B i , so for any complex number Z we have the equality =Z.

module complex number Z=A+B i called number and is denoted by , i.e.

From formula (1) it follows that for any complex number Z, and =0 if and only if Z=0, i.e. when A=0 and B=0. Let us prove that for any complex number Z the formulas are valid:

4. ADDITION AND MULTIPLICATION OF COMPLEX NUMBERS

sum two complex numbers A+B i and C+D i is called a complex number (A+C ) + ( B+D)· i , i.e. ( A+B i) + ( C+D i)=( A+C) + (B+D) i

work two complex numbers A+B i and C+D i is called a complex number (A C – B D)+(A D+B C) i , i.e.

(A + B i ) (C + D i )=(A C – B D) + (A D + B C) i

Transfer property:

Z 1 + Z 2 \u003d Z 2 + Z 1, Z 1 Z 2 \u003d Z 2 Z 1

Associative property:

(Z 1 + Z 2) + Z 3 \u003d Z 1 + (Z 2 + Z 3), (Z 1 Z 2) Z 3 \u003d Z 1 (Z 2 Z 3)

Distribution property:

Z 1 (Z 2 + Z 3) \u003d Z 1 Z 2 + Z 1 Z 3

Geometric representation of the sum of complex numbers

According to the definition of addition of two complex numbers, the real part of the sum is equal to the sum of the real parts of the terms, the imaginary part of the sum is equal to the sum of the imaginary parts of the terms. The coordinates of the sum of vectors are determined in the same way:

The sum of two vectors with coordinates (A 1 ;B 1) and (A 2 ;B 2) is a vector with coordinates (A 1 +A 2 ;B 1 +B 2). Therefore, to find the vector corresponding to the sum of the complex numbers Z 1 and Z 2, you need to add the vectors corresponding to the complex numbers Z 1 and Z 2 .

Example 1: Find the sum and product of complex numbers Z 1 =2 - 3× i and

Z 2 \u003d -7 + 8 × i .

Z 1 + Z 2 \u003d 2 - 7 + (-3 + 8) × i = - 5 + 5× i

Z 1× Z 2 = (2 – 3× i )× (–7 + 8× i ) = –14 + 16× i + 21× i + 24 = 10 + 37× i

5. SUBTRACTION AND DIVISION OF COMPLEX NUMBERS

Subtraction of complex numbers is the inverse operation of addition: for any complex numbers Z 1 and Z 2 there exists, and moreover, only one, the number Z, such that:

If we add to both parts of the equality (–Z 2) the opposite of the number Z 2:

Z + Z 2 + (-Z 2) \u003d Z 1 + (-Z 2), whence

The number Z \u003d Z 1 + Z 2 is called difference of numbers Z 1 and Z 2 .

Division is introduced as the inverse of multiplication:

Z × Z 2 \u003d Z 1

Dividing both parts by Z 2 we get:

This equation shows that Z 2 0

Geometric representation of the difference of complex numbers

The difference Z 2 - Z 1 of complex numbers Z 1 and Z 2 corresponds to the difference of the vectors corresponding to the numbers Z 1 and Z 2 . The module of the difference of two complex numbers Z 2 and Z 1 by definition of the module is the length of the vector Z 2 - Z 1 . We construct this vector as the sum of the vectors Z 2 and (–Z 1) (Figure 4). Thus, the modulus of the difference of two complex numbers is the distance between the points of the complex plane that correspond to these numbers.

This important geometric interpretation of the modulus of the difference of two complex numbers makes it possible to use simple geometric facts with success.

Example 2: Given complex numbers Z 1 = 4 + 5 i and Z 2 = 3 + 4 i . Find the difference Z 2 - Z 1 and the quotient

Z 2 - Z 1 \u003d (3 + 4 ) i) – (4 + 5 i) = –1 – i

6. TRIGONOMETRIC FORM OF A COMPLEX NUMBER

Writing a complex number Z as A+B i called algebraic form complex number. In addition to the algebraic form, other forms of writing complex numbers are also used.

Consider trigonometric form notation of a complex number. Real and imaginary parts of a complex number Z=A+B i are expressed in terms of its modulus = r and argument j as follows:

A= r cosj ; B= r sinj .

The number Z can be written like this:

Z= r cosj + i sinj = r (cosj + i sinj)

Z = r (cosj + i sinj ) (2)

This entry is called trigonometric form of a complex number .

r = is the modulus of the complex number.

The number j is called complex number argument.

The argument of the complex number Z0 is the value of the angle between the positive direction of the real axis and the vector Z, and the value of the angle is considered positive if the count is counterclockwise, and negative if it is clockwise.

For the number Z=0, the argument is not defined, and only in this case the number is given only by its modulus.

As mentioned above = r =, equality (2) can be written as

A+B i =· cosj + i · sinj , whence, equating the real and imaginary parts, we obtain:

cosj =, sinj = (3)

If sinj divide by cosj we get:

tgj= (4)

This formula is more convenient to use to find the argument j than formulas (3). However, not all values of j that satisfy equality (4) are arguments of the number A + B i . Therefore, when finding the argument, you need to take into account in which quarter the point A + B is located i .

7. PROPERTIES OF THE MODULUS AND ARGUMENT OF A COMPLEX NUMBER

Using the trigonometric form, it is convenient to find the product and the quotient of complex numbers.

Let Z 1 = r 1 ( cosj 1 +i sinj 1), Z 2 = r 2 ( cosj 2 +i sinj 2). Then:

Z 1 Z 2 = r 1 r 2 =

= r 1 r 2 .

Thus, the product of complex numbers written in trigonometric form can be found by the formula:

Z 1 Z 2 = r 1 r 2 (5)

From formula (5) it follows that when complex numbers are multiplied, their moduli are multiplied and the arguments are added.

If Z 1 \u003d Z 2 then we get:

Z2=2= r 2 (cos2j +i sin2j)

Z 3 \u003d Z 2 Z \u003d r 2 ( cos2j +i sin2j ) r (cosj + i sinj )=

= r 3 ( cos3j +i sin3j)

In general, for any complex number Z=r (cosj + i sinj )0 and any natural number n, the formula is valid:

Zn=[ r (cosj + i sinj )] n = r n (cosnj + i sinnj ),(6)

which is called De Moivre's formula.

The quotient of two complex numbers written in trigonometric form can be found by the formula:

[cos(j 1 – j 2) + i sin(j 1 - j 2)].(7)

= = cos(–j 2) + i sin(–j 2)

Using formula 5

(cosj 1 + i sinj 1)× (cos(–j 2) + i sin(–j 2)) =

cos(j 1 – j 2) + i sin(j 1 - j 2).

Example 3:

We write the number -8 in trigonometric form

8 = 8 (cos(p + 2p k ) + i sin(p + 2p k )), k О Z

Let Z = r× (cosj + i×

r 3× (cos3j + i× sin3j ) = 8 (cos(p + 2p k ) + i sin(p + 2p k )), k О Z

Then 3j =p + 2p k , k О Z

j= , k О Z

Hence:

Z = 2 (cos() + i sin()), k О Z

k = 0,1,2...

k = 0

Z 1 = 2 (cos + i sin) = 2 ( i) = 1+× i

k = 1

Z 2 = 2 (cos( + ) + i sin( + )) = 2 (cosp + i sinp) = –2

k = 2

Z 3 = 2 (cos( + ) + i sin( + )) = 2 (cos + i sin) = 1–× i

Answer: Z 13 =; Z 2 \u003d -2

Example 4:

We write the number 1 in trigonometric form

1 = 1 (cos(2p k ) + i sin(2p k )), k О Z

Let Z = r× (cosj + i× sinj ), then this equation will be written as:

r 4× (cos4j + i× sin4j ) = cos(2p k ) + i sin(2p k )), k О Z

4j = 2p k , k О Z

j = , k О Z

Z = cos + i× sin

k = 0,1,2,3...

k = 0

Z1 = cos0+ i× sin0 = 1 + 0 = 1

k = 1

Z2 = cos+ i× sin=0+ i = i

k = 2

Z 3 \u003d cosp + i sinp = -1 + 0 = -1

k = 3

Z4 = cos+ i× sin

Answer: Z 13 = 1

Z 24 = i

8. Exponentiation and Extraction of the Root

Formula 6 shows that raising the complex number r (cosj + i sinj ) to a positive integer power with a natural exponent, its modulus is raised to a power with the same exponent, and the argument is multiplied by the exponent.

[ r (cosj + i sinj )] n = r n (cos nj + i sin nj )

Number Z called degree root n from the number w (denoted by ) if Z n =w .

From this definition it follows that each solution of the equation Z n = w is the root of the degree n from number w. In other words, in order to extract the root of the degree n from the number w, it is enough to solve the equation Z n =w. If w =0, then for any n the equation Z n = w has only one solution Z= 0. If w 0, then Z0 , and, consequently, Z and w can be represented in trigonometric form

Z = r (cosj + i sinj ), w = p (cozy + i siny)

The equation Z n = w will take the form:

r n (cos nj + i sin nj ) = p (cozy + i siny)

Two complex numbers are equal if and only if their moduli are equal and their arguments differ in terms that are multiples of 2p . Therefore, r n = p and nj = y + 2p k , where kн Z or r = and j= , where kн Z .

So, all solutions can be written as follows:

Z K =, kн Z (8)

Formula 8 is called De Moivre's second formula.

Thus, if w 0, then there are exactly n roots of degree n from the number w: all of them are contained in formula 8. All roots of degree n from the number w have the same modulus , but different arguments differing in the summand that is a multiple of the number . It follows that the complex numbers that are roots of degree n from the complex number w correspond to the points of the complex plane located at the vertices of a regular n-gon inscribed in a circle of radius centered at the point Z = 0.

The symbol does not have an unambiguous meaning. Therefore, when using it, one should clearly understand what is meant by this symbol. For example, using the notation , you should think about making it clear that this symbol means a pair of complex numbers i and –i , or one, then which one.

Equations of higher powers

Formula 8 defines all roots of a two-term equation of degree n. The situation is immeasurably more complicated in the case of a general algebraic equation of degree n:

a n × Z n+ a n–1× Z n–1 +...+ a 1× Z 1 + a 0 = 0(9)

Where a n ,..., a 0 are given complex numbers.

In the course of higher mathematics, Gauss's theorem is proved: every algebraic equation has at least one root in the set of complex numbers. This theorem was proved by the German mathematician Carl Gauss in 1779.

Based on the Gauss theorem, we can prove that the left side of Equation 9 can always be represented as a product:

Where Z 1 , Z 2 ,..., Z K are some different complex numbers,

and a 1 ,a 2 ,...,a k are natural numbers, and:

a 1 + a 2 + ... + a k = n

This implies that the numbers Z 1 , Z 2 ,..., Z K are the roots of equation 9. In this case, they say that Z 1 is a root of multiplicity a 1 , Z 2 is a root of multiplicity a 2 and so on.

Gauss's theorem and the theorem just formulated give solutions about the existence of roots, but say nothing about how to find these roots. If the roots of the first and second degrees can be easily found, then for the equations of the third and fourth powers the formulas are cumbersome, and for equations of a degree higher than the fourth such formulas do not exist at all. The absence of a general method does not prevent finding all the roots of the equation. To solve an equation with integer coefficients, the following theorem is often useful: the integer roots of any algebraic equation with integer coefficients are divisors of the constant term.

Let's prove this theorem:

Let Z = k be an integer root of the equation

a n× Z n + a n–1× Z n–1 +...+ a 1× Z 1 + a 0 = 0

with integer coefficients. Then

a n× k n + a n–1× k n–1 +...+ a 1× k 1 + a 0 = 0

a 0 = – k(a n× k n–1 + a n–1× k n–2 +...+ a 1)

The number in brackets, under the assumptions made, is obviously an integer, which means that k is a divisor of the number a 0 .

9.QUADRATIC EQUATION WITH A COMPLEX UNKNOWN

Consider the equation Z 2 = a, where a is a given real number, Z is an unknown.

This is the equation:

We write the number a in the form a = (– 1)× (– a) = i 2× = i 2× () 2 . Then the equation Z 2 = a will be written in the form: Z 2 - i 2×() 2 = 0

those. (Z- i× )(Z + i× ) = 0

Therefore, the equation has two roots: Z 1.2 = i×

The introduced concept of a root from a negative number allows us to write down the roots of any quadratic equation with real coefficients

a× Z 2 + b× Z + c = 0

According to the well-known general formula

Z 1,2 = (10)

So, for any real a(a0), b, c, the roots of the equation can be found by formula 10. Moreover, if the discriminant, i.e. radical expression in formula 10

D \u003d b 2 - 4 × a × c

is positive, then the equation a× Z 2 + b× Z + c = 0 has two real distinct roots. If D = 0, then the equation a × Z 2 + b × Z + c = 0 has one root. If D< 0, то уравнение a× Z 2 + b× Z + c = 0 имеет два различных комплексных корня.

The complex roots of a quadratic equation have the same properties as the known properties of real roots.

Let's formulate the main ones:

Let Z 1 ,Z 2 be the roots of the quadratic equation a× Z 2 + b× Z + c = 0, a0. Then the properties are true:

Z 1× Z 2 =

For all complex Z, the formula

a × Z 2 + b × Z + c \u003d a × (Z - Z 1) × (Z - Z 2)

Example 5:

Z 2 - 6 Z + 10 \u003d 0

D \u003d b 2 - 4 a c

D \u003d 6 2 - 4 10 \u003d - 4

– 4 = i 2 ·4

Z 1,2 =

Answer: Z 1 \u003d Z 2 \u003d 3 + i

Example 6:

3 Z 2 +2 Z + 1 = 0

D \u003d b 2 - 4 a c

D \u003d 4 - 12 \u003d - 8

D \u003d -1 8 \u003d 8 i 2

Z 1,2 = =

Answer: Z 1 \u003d Z 2 \u003d -

Example 7:

Z 4 - 8 Z 2 - 9 = 0

t 2 - 8 t - 9 = 0

D \u003d b 2 - 4 a c \u003d 64 + 36 \u003d 100

t 1 \u003d 9 t 2 \u003d - 1

Z 2 \u003d 9 Z 2 \u003d - 1

Z 3.4 = i

Answer: Z 1.2 \u003d 3, Z 3.4 \u003d i

Example 8:

Z 4 + 2 Z 2 - 15 \u003d 0

t 2 + 2 t - 15 \u003d 0

D \u003d b 2 - 4 a c \u003d 4 + 60 \u003d 64

t 1,2 = = = –14

t 1 \u003d - 5 t 2 \u003d 3

Z 2 \u003d - 5 Z 2 \u003d 3

Z 2 \u003d - 1 5 Z 3.4 \u003d

Z2 = i 2 ·five

Z 1,2 = i

Answer: Z 1,2 = i , Z 3,4 =

Example 9:

Z 2 \u003d 24 - 10 i

Let Z = X + Y i

(X + Y i ) 2 = X 2 + 2 X Y i  Y2

X 2 + 2 X Y Y i  Y 2 \u003d 24 - 10 i

(X 2 - Y 2) + 2 X Y i = 24 – 10 i

multiply by X 2 0

X 4 - 24 X 2 - 25 = 0

t 2 - 24 t - 25 = 0

t 1 t 2 \u003d - 25

t 1 \u003d 25 t 2 \u003d - 1

X 2 \u003d 25 X 2 \u003d - 1 - no solutions

X 1 \u003d 5 X 2 \u003d - 5

Y 1 = - Y 2 =

Y 1 = - 1 Y 2 = 1

Z 1,2 \u003d (5 - i )

Answer: Z 1,2 \u003d (5 - i )

TASKS:

(2 - Y) 2 + 3 (2 - Y) Y + Y 2 = 6

4 - 4 Y + Y 2 + 6 Y - 3 Y 2 + Y 2 = 6

-Y 2 + 2Y - 2 \u003d 0 / -1

Y 2 - 2Y + 2 = 0

D \u003d b 2 - 4 a c \u003d 4 - 8 \u003d - 4

– 4 = – 1 4 = 4 i 2

Y 1,2 = = = 1 i

Y 1 = 1– i Y 2 \u003d 1 + i

X 1 = 1 + i X 2 \u003d 1– i

Answer: (1 + i ; 1–i }

{1–i ; 1 + i }

Let's square it

§one. Complex numbers

1°. Definition. Algebraic notation.

Definition 1. Complex numbers called ordered pairs of real numbers and , if the concept of equality is defined for them, the operations of addition and multiplication that satisfy the following axioms:

1) Two numbers
and
equal if and only if
,
, i.e.


,
.

2) The sum of complex numbers
and

and equal
, i.e.

+
=
.

3) The product of complex numbers
and
the number is called
and equal to , i.e.

∙=.

The set of complex numbers is denoted C.

Formulas (2),(3) for numbers of the form
take the form

whence it follows that the operations of addition and multiplication for numbers of the form
coincide with addition and multiplication for real numbers  a complex number of the form
is identified with a real number .

Complex number
called imaginary unit and denoted , i.e.
Then from (3) 

From (2),(3)  which means

Expression (4) is called algebraic notation complex number.

In algebraic form, the operations of addition and multiplication take the form:

The complex number is denoted
, - the real part, is the imaginary part, is a purely imaginary number. Designation:
,
.

Definition 2. Complex number
called conjugate with a complex number
.

Properties of complex conjugation.

2)
.

3) If
, then
.

4)
.

5)
is a real number.

The proof is carried out by direct calculation.

Definition 3. Number
called module complex number
and denoted
.

It's obvious that
, and

. The formulas are also obvious:
and
.

2°. Properties of addition and multiplication operations.

1) Commutativity:
,
.

2) Associativity:,
.

3) Distributivity: .

The proof 1) - 3) is carried out by direct calculations based on similar properties for real numbers.

4)
,
.

5) , C ! , satisfying the equation
. Such

6) ,C, 0, ! :
. Such is found by multiplying the equation by

.

Example. Imagine a complex number
in algebraic form. To do this, multiply the numerator and denominator of the fraction by the conjugate of the denominator. We have:

3°. Geometric interpretation of complex numbers. Trigonometric and exponential form of writing a complex number.

Let a rectangular coordinate system be given on the plane. Then
C one can associate a point on the plane with coordinates
.(see Fig. 1). It is obvious that such a correspondence is one-to-one. In this case, real numbers lie on the abscissa axis, and purely imaginary numbers lie on the ordinate axis. Therefore, the abscissa axis is called real axis, and the y-axis − imaginary axis. The plane on which the complex numbers lie is called complex plane.

Note that and
are symmetrical about the origin, and and are symmetrical with respect to Ox.

Each complex number (i.e., each point on the plane) can be associated with a vector starting at the point O and ending at the point
. The correspondence between vectors and complex numbers is one-to-one. Therefore, the vector corresponding to the complex number , denoted by the same letter

D vector line
corresponding to the complex number
, is equal to
, and
,
.

Using the vector interpretation, one can see that the vector
− sum of vectors and , a
− sum of vectors and
.(see Fig. 2). Therefore, the following inequalities are true:

Along with the length vector we introduce the angle between vector and the Ox axis, counted from the positive direction of the Ox axis: if the count is counterclockwise, then the sign of the angle is considered positive, if clockwise, then negative. This corner is called complex number argument and denoted
. Injection is not defined uniquely, but with precision
…. For
the argument is not defined.

Formulas (6) define the so-called trigonometric notation complex number.

From (5) it follows that if
and
then

,
.

From (5)
what by and A complex number is uniquely defined. The converse is not true: namely, by the complex number its module is unique, and the argument , due to (7), − with accuracy
. It also follows from (7) that the argument can be found as a solution to the equation

However, not all solutions to this equation are solutions to (7).

Among all the values of the argument of a complex number, one is chosen, which is called the main value of the argument and is denoted
. Usually the main value of the argument is chosen either in the interval
, or in the interval

In trigonometric form, it is convenient to perform multiplication and division operations.

Theorem 1. Module of the product of complex numbers and is equal to the product of the modules, and the argument is equal to the sum of the arguments, i.e.

, but .

Similarly

Proof. Let be , . Then by direct multiplication we get:

Similarly

.■

Consequence(De Moivre's formula). For
Moivre's formula is valid

P example. Let Find the geometric location of the point
. It follows from Theorem 1 that .

Therefore, to construct it, you must first construct a point , which is the inverse about the unit circle, and then find a point symmetrical to it about the x-axis.

Let
, i.e.
Complex number
denoted
, i.e. R the Euler formula is valid

Because
, then
,
. From Theorem 1
what about the function
it is possible to work as with an ordinary exponential function, i.e. equalities are true

,
,
.

From (8)
exponential notation complex number

, where
,

Example. .

4°. Roots th power of a complex number.

Consider the equation

,
WITH ,
N .

Let
, and the solution of Eq. (9) is sought in the form
. Then (9) takes the form
, whence we find that
,
, i.e.

,
,
.

Thus, equation (9) has roots

,
.

Let us show that among (10) there are exactly various roots. Really,

are different, because their arguments are different and differ less than
. Further,
, because
. Similarly
.

Thus, equation (9) for
has exactly roots
located at the vertices of a regular -gon inscribed in a circle of radius centered at T. O.

Thus, it has been proven

Theorem 2. root extraction th power of a complex number
always possible. All root values th degree of located at the top of the correct -gon inscribed in a circle with center at zero and radius
. Wherein,

Consequence. Roots -th degree of 1 are expressed by the formula

The product of two roots of 1 is a root, 1 is a root -th degree from unity, root
:
.

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, the symbol whose square is -1, i.e. i 2 = -1. Number a called real part, and the number b - imaginary part complex number z = a + bi. If b= 0, then instead of a + 0i write simply 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 ones: they can be added, subtracted, multiplied and divided by each other. Addition and subtraction proceed according to the rule ( a + bi) ± ( c + di) = (a ± c) + (b ± d)i, and multiplication - according to the rule ( a + bi) · ( c + di) = (ac – bd) + (ad + bc)i(here it is just 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: the number z = a + bi can be represented as a vector with coordinates ( a; b) on the Cartesian plane (or, which is almost the same, a point - the end of the 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 by the parallelogram rule). By the Pythagorean theorem, the length of the vector with coordinates ( a; b) is equal to . This value 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 denoted by Arg z. The argument is not uniquely defined, but only up to the addition of a multiple of 2 π radians (or 360°, if you count in degrees) - after all, it is clear that turning through 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 ). Hence 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 calculations. Multiplication of complex numbers in trigonometric form looks very simple: z one · 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 moduli are multiplied and the arguments are added). From here follow De Moivre formulas: z n = |z|n(cos( n(Arg z)) + i sin( n(Arg z))). With the help of these formulas, it is easy to learn how to extract roots of any degree from complex numbers. nth root of z is such 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- one). This means that there is always exactly n roots n th degree from a complex number (on the plane they are located at the vertices of a regular n-gon).

TopicComplex numbers and polynomials

Lecture 22

§one. Complex numbers: basic definitions

Symbol enter the ratio
and is called the imaginary unit. In other words,
.

Definition. Expression of the form
, where
, is called a complex number, and the number called the real part of a complex number and denote
, number - imaginary part and denote
.

From this definition it follows that the real numbers are those complex numbers whose imaginary part is equal to zero.

It is convenient to represent complex numbers as points of a plane on which a Cartesian rectangular coordinate system is given, namely: a complex number
match point
and vice versa. on axle
real numbers are displayed and it is called the real axis. Complex numbers of the form

are called purely imaginary. They are shown as dots on the axis.
, which is called the imaginary axis. This plane, which serves to represent complex numbers, is called the complex plane. A complex number that is not real, i.e. such that
, sometimes called imaginary.

Two complex numbers are said to be equal if and only if they have the same real and imaginary parts.

Addition, subtraction and multiplication of complex numbers are performed according to the usual rules of polynomial algebra, taking into account the fact that

. The division operation can be defined as the inverse of the multiplication operation and one can prove the uniqueness of the result (if the divisor is different from zero). However, in practice, a different approach is used.

Complex numbers
and
are called conjugate, on the complex plane they are represented by points symmetric about the real axis. It's obvious that:

1)

;

2)
;

3)
.

Now split on the can be done as follows:

It is not difficult to show that

where symbol stands for any arithmetic operation.

Let
some imaginary number, and is a real variable. The product of two binomials

is a square trinomial with real coefficients.

Now, having complex numbers at our disposal, we can solve any quadratic equation
.If , then

and the equation has two complex conjugate roots

If
, then the equation has two different real roots. If
, then the equation has two identical roots.

§2. Trigonometric form of a complex number

As mentioned above, the complex number
convenient to represent with a dot
. One can also identify such a number with the radius vector of this point
. With this interpretation, the addition and subtraction of complex numbers is performed according to the rules of addition and subtraction of vectors. For multiplication and division of complex numbers, another form is more convenient.

We introduce on the complex plane
polar coordinate system. Then where
,
and complex number
can be written as:

This form of notation is called trigonometric (in contrast to the algebraic form
). In this form, the number is called a module and - complex number argument . They are marked:
,

. For the module, we have the formula

The number argument is defined ambiguously, but up to a term
,
. The value of the argument that satisfies the inequalities
, is called principal and denoted
. Then,
. For the main value of the argument, you can get the following expressions:

number argument
considered to be undefined.

The condition for the equality of two complex numbers in trigonometric form has the form: the modules of the numbers are equal, and the arguments differ by a multiple
.

Find the product of two complex numbers in trigonometric form:

So, when multiplying numbers, their modules are multiplied, and the arguments are added.

Similarly, it can be established that when dividing, the modules of numbers are divided, and the arguments are subtracted.

Understanding exponentiation as multiple multiplication, we can obtain the formula for raising a complex number to a power:

We derive a formula for
- root th power of a complex number (not to be confused with the arithmetic root of a real number!). The root extraction operation is the inverse of the exponentiation operation. So
is a complex number such that
.

Let
known, and
required to be found. Then

From the equality of two complex numbers in trigonometric form, it follows that

,
,
.

From here
(it's an arithmetic root!),

,
.

It is easy to verify that can only accept essentially different values, for example, when
. Finally we have the formula:

,
.

So the root th degree from a complex number has different values. On the complex plane, these values \u200b\u200bare located at the vertices correctly -gon inscribed in a circle of radius
centered at the origin. The “first” root has an argument
, the arguments of two “neighboring” roots differ by
.

Example. Let's take the cube root of the imaginary unit:
,
,
. Then: