§1. Complex numbers

1°. Definition. Algebraic notation.

Definition 1. Complex numbers ordered pairs of real numbers are called And , if for them the concept of equality, addition and multiplication operations are defined, satisfying 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) Product of complex numbers
And
is the number denoted by
and equal, 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  complex number of the form
identified with a real number .

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

From (2), (3)  which means

Expression (4) is called algebraic notation complex number.

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

A complex number is denoted by
, – real part, – imaginary part, is a purely imaginary number. Designation:
,
.

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

Properties of complex conjugation.

1)

2)
.

3) If
, That
.

4)
.

5)
– real number.

The proof is carried out by direct calculation.

Definition 3. Number
called module complex number
and is designated
.

It's obvious that
, and


. The formulas are also obvious:
And
.

2°. Properties of addition and multiplication operations.

1) Commutativity:
,
.

2) Associativity:,
.

3) Distributivity: .

Proof 1) – 3) is carried out by direct calculations based on similar properties for real numbers.

4)
,
.

5) , C ! , satisfying the equation
. This

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



.

Example. Let's imagine a complex number
in algebraic form. To do this, multiply the numerator and denominator of the fraction by the conjugate number 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 specified on the plane. Then
C you can match a point on the plane with the coordinates
.(see Fig. 1). Obviously, 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 ordinate axis − imaginary axis. The plane on which complex numbers lie is called complex plane.

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

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

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

Using vector interpretation, we can see that the vector
− sum of vectors And , A
− sum of vectors And
.(see Fig. 2). Therefore, the following inequalities are valid: ,

Along with the length vector let's introduce the angle between vector and the Ox axis, counted from the positive direction of the Ox axis: if the counting is counterclockwise, then the sign of the angle is considered positive, if clockwise, then negative. This angle is called complex number argument and is designated
. Corner is not determined unambiguously, 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
That

, .

From (5)
what about And a complex number is uniquely determined. The converse is not true: namely, over a complex number its module is unique, and the argument , by virtue of (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 selected, 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

It is convenient to perform multiplication and division operations in trigonometric form.

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

, A .

Likewise

,

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

Likewise

.■

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

P example. Let us find the geometric location of the point
. From Theorem 1 it follows that .

Therefore, to construct it, you must first construct a point , which is the inversion relative to the unit circle, and then find a point symmetrical to it relative to the Ox axis.

Let
, i.e.
Complex number
denoted by
, i.e. R Euler's formula is valid

Because
, That
,
. From Theorem 1
what's with the function
you can work as with a regular exponential function, i.e. equalities are valid

,
,
.

From (8)
demonstrative notation complex number

, Where
,

Example. .

4°. Roots -th power of a complex number.

Consider the equation

, WITH , N .

Let
, and the solution to equation (9) is sought in the form
. Then (9) takes the form
, from where we find that
,
, i.e.

,
,
.

Thus, equation (9) has roots

, .

Let us show that among (10) there is exactly different roots. Really,

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

Thus, equation (9) at
has exactly roots
, located at the vertices of the regular -a triangle inscribed in a circle of radius with center at point O.

Thus it is proven

Theorem 2. Root extraction -th power of a complex number
It's always possible. All root meanings th degree of located at the vertices of the correct -gon inscribed in a circle with center at zero and radius
. Wherein,

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

.

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

Complex numbers. History of discovery

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

F. Klein

Ancient Greek mathematicians considered only natural numbers to be “real”. Gradually, the idea of ​​the infinity of the set of natural numbers took shape.

In the 3rd century, Archimedes developed a system of notation up to such a huge one as

. Along with natural numbers, fractions were used - numbers made up of a whole number of fractions of a unit. Fractions were used in practical calculations two thousand years BC. e. V ancient Egypt and ancient Babylon. For a long time it was believed that the result of a measurement is always expressed either as a natural number or as a ratio of such numbers, that is, a fraction. The ancient Greek philosopher and mathematician Pythagoras taught that "... the elements of numbers are the elements of all things, and the whole world as a whole is harmony and number." The strongest blow to this view was dealt by a discovery made by one of the Pythagoreans. He proved that the diagonal of a square is incommensurate with its side. It follows that natural numbers and fractions are not enough to express the length of the diagonal of a square with side 1. There is reason to assert that it was with this discovery that the era of theoretical mathematics began: to discover the existence of incommensurable quantities with the help of experience, without resorting to abstract reasoning, was impossible.

Next important stage in the development of the concept of number there was the introduction of negative numbers - this was done by Chinese mathematicians two centuries BC. e. Negative numbers were used in the 3rd century by the ancient Greek mathematician Diophantus, who already knew the rules for operating on them, and in the 7th century these numbers were already studied in detail by Indian scientists who compared such numbers with debt. With the help of negative numbers it was possible to describe changes in quantities in a unified way. Already in the 8th century it was established that the square root of a positive number has two meanings - positive and negative, and the square root cannot be taken from negative numbers: there is no such number

, to .

In the 16th century, in connection with the study of cubic equations, it became necessary to extract square roots of negative numbers. In the formula for solving cubic equations of the form

cubic and square roots: .

This formula works flawlessly in the case when the equation has one real root (

), and if it has three real roots ( ), then under the square root sign there was a negative number. It turned out that the path to these roots leads through the impossible operation of extracting the square root of a negative number. After the 4th degree equations were solved, mathematicians intensively searched for a formula to solve the 5th degree equation. But Ruffini (Italy) at the turn of the 18th and 19th centuries proved that a letter equation of the fifth degree cannot be solved algebraically; more precisely, it is impossible to express its root through the literal quantities a, b, c, d, e using six algebraic operations (addition, subtraction, multiplication, division, exponentiation, root extraction).

In 1830, Galois (France) proved that no general equation whose degree is greater than 4 can be solved algebraically. However, every equation nth degree has (if we also consider complex numbers) n roots (among which there may be equal ones). Mathematicians were convinced of this back in the 17th century (based on the analysis of numerous special cases), but only at the turn of the 18th and 19th centuries the mentioned theorem was proven by Gauss.

The Italian algebraist G. Cardano in 1545 proposed introducing numbers of a new nature. He showed that a system of equations that has no solutions in the set of real numbers has solutions of the form

, , you just need to agree to act on such expressions according to the rules of ordinary algebra and assume that . Cardano called such quantities " purely negative" and even " sophistically negative", considered them useless and tried not to use them. In fact, with the help of such numbers it is impossible to express either the result of measuring any quantity, or the change in any quantity. But already in 1572, a book by the Italian algebraist R. Bombelli was published, in which established the first rules for arithmetic operations on such numbers, up to the extraction of cube roots from them. Title " imaginary numbers"was introduced in 1637 by the French mathematician and philosopher R. Descartes, and in 1777, one of the greatest mathematicians of the 18th century, L. Euler, proposed using the first letter French word imagine(imaginary) to denote a number (imaginary unit). 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 Latin complexus) means a connection, combination, a set of concepts, objects, phenomena, etc., forming a single whole.

During the 17th century, discussions continued about the arithmetic nature of imaginary numbers and the possibility of giving them a geometric justification.

The technique of operations on imaginary numbers gradually developed. At the turn of the 17th and 18th centuries, a general theory of nth roots was constructed, first from negative and then from any complex numbers, based on the following formula of the English mathematician A. Moivre (1707):

. Using this formula, it was also possible to derive formulas for the cosines and sines of multiple arcs. L. Euler derived a remarkable formula in 1748: , which linked together the exponential function with the trigonometric one. Using L. Euler's formula, it was possible to raise the number e to any complex power. It is interesting, for example, that . You can find sin and cos from complex numbers, calculate the logarithms of such numbers, that is, build a theory of functions of a complex variable.

At the end of the 18th century, the French mathematician J. Lagrange was able to say that mathematical analysis was no longer complicated by imaginary quantities. Using imaginary numbers, we learned to express solutions to linear differential equations with constant coefficients. Such equations are found, for example, in the theory of oscillations of a material point in a resisting medium. Even earlier, the Swiss mathematician J. Bernoulli used complex numbers to solve integrals.

Although during the 18th century many questions were solved with the help of complex numbers, including applied problems related to cartography, hydrodynamics, etc., there was still no strictly logical justification 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 induction, acquiring the character of real truths only after confirmation by direct evidence.

“No one doubts the accuracy of the results obtained from calculations with imaginary quantities, although they are only algebraic forms of hieroglyphs of absurd quantities,” wrote L. Carnot.

At the end of the 18th century, at the beginning of the 19th century, a geometric interpretation of complex numbers was obtained. The Dane K. Wessel, the Frenchman J. Argan and the German K. Gauss independently proposed to depict a complex number

point on the coordinate plane. Later it turned out that it is even more convenient to represent a number not as a point itself M, and by vector

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 the real numbers are special case 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 multiple of 2 π radians (or 360°, if counted in degrees) - after all, it is clear that a rotation 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 powers from number 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).

SubjectComplex numbers and polynomials

Lecture 22

§1. Complex numbers: basic definitions

Symbol is introduced by 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 real numbers are those complex numbers whose imaginary part is equal to zero.

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

are called purely imaginary. They are represented by points 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 both their real and imaginary parts are the same.

Addition, subtraction and multiplication of complex numbers is carried out 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 the uniqueness of the result can be proven (if the divisor is non-zero). However, in practice a different approach is used.

Complex numbers
And
are called conjugate; on the complex plane they are represented by points symmetrical about the real axis. It's obvious that:

1)

;

2)
;

3)
.

Now split on can be done as follows:

.

It's not difficult to show that

,

where is the symbol stands for any arithmetic operation.

Let
some imaginary number, and – real variable. Product of two binomials

is a quadratic 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, a complex number
convenient to represent as a dot
. This number can also be identified with the radius vector of this point
. With this interpretation, addition and subtraction of complex numbers is carried out according to the rules for addition and subtraction of vectors. For multiplying and dividing complex numbers, another form is more convenient.

Let us 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 – argument of a complex number . They are designated:
,

. For the module we have the formula

The argument of a number is not uniquely defined, but up to a term
,
. The value of the argument satisfying the inequalities
, is called the main one and is denoted
. Then,
. For the main value of the argument, you can get the following expressions:

,

number argument
is considered uncertain.

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 of
.

Let's find the product of two complex numbers in trigonometric form:

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

In a similar way, we can establish that when dividing, the modules of numbers are divided and the arguments are subtracted.

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

Let us derive a formula for
– root -th power of a complex number (not to be confused with the arithmetic root of a real number!). The operation of extracting the root is the inverse of the operation of exponentiation. That's why
is a complex number such that
.

Let
is known, but
required to be found. Then

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

,
,
.

From here
(this is 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 the th power of a complex number has different meanings. On the complex plane, these values ​​are located correctly at the vertices -a triangle inscribed in a circle of radius
with center 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:

,

HISTORICAL REFERENCE

Complex numbers were introduced into mathematics to make it possible to take the square root of any real number. This, however, is not a sufficient reason to introduce new numbers into mathematics. It turned out that if you carry out calculations according to the usual rules on expressions in which the square root of a negative number occurs, you can come to a result that no longer contains the square root of a negative number. In the 16th century Cardano found a formula for solving the cubic equation. It turned out that when a cubic equation has three real roots, the Cardano formula contains the square root of a negative number. Therefore, square roots of negative numbers began to be used in mathematics and were called imaginary numbers - thereby, as it were, they acquired the right to 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. CONCEPT OF COMPLEX NUMBER

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

So, to solve 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.

Algebraic equations of the first degree are solvable on the set of rational numbers, i.e. equations of the form A· X+B=0 (A0). However, algebraic equations of degree higher than first may not have rational roots. For example, these 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, we have 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 a set complex numbers.

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

As for real numbers, it is necessary to introduce the operations of addition and multiplication of complex numbers so that their sum and product are complex numbers. Then, in particular, for any real numbers A and B the expression A+B+ i can be considered as writing a complex number in general view. 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 – some symbol such that i 2 = –1, and is 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 called the imaginary unit.

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

To strictly define 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 are 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 COMPLEX NUMBER

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

O(0;0) and with the end at 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. COMPLEX NUMBER MODULE

Let a complex number Z=A+B· be given i . Conjugate With Z is called a complex number A – B i , which is denoted, i.e.

A – B i .

Note that = A+B· i , therefore for any complex number Z the equality =Z holds.

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 following formulas are valid:

4.ADDING AND MULTIPLYING COMPLEX NUMBERS

Amount 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

The 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

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

Displacement property:

Z 1 +Z 2 =Z 2 +Z 1, Z 1· Z 2 =Z 2· Z 1

Matching property:

(Z 1 +Z 2)+Z 3 =Z 1 +(Z 2 +Z 3), (Z 1 Z 2) Z 3 =Z 1 (Z 2 Z 3)

Distribution property:

Z 1 (Z 2 +Z 3)=Z 1 Z 2 +Z 1 Z 3

Geometric representation of the sum of complex numbers

According to the definition of adding 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 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 = –7 + 8× i .

Z 1 + Z 2 = 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.SUBTRACT 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 is, and only one, the number Z, such that:

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

Z+Z 2 +(–Z 2)=Z 1 +(–Z 2), whence

The number Z=Z 1 +Z 2 is called difference in numbers Z 1 and Z 2.

Division is introduced as the inverse operation of multiplication:

Z×Z 2 =Z 1

Dividing both sides by Z 2 we get:

From this equation it is clear that Z 2 0

Geometric representation of the difference of complex numbers

The difference Z 2 – Z 1 of the complex numbers Z 1 and Z 2 corresponds to the difference of the vectors corresponding to the numbers Z 1 and Z 2. The modulus of the difference between two complex numbers Z 2 and Z 1, by definition of the modulus, is the length of the vector Z 2 – Z 1. Let's 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 simple geometric facts useful.

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 = (3 + 4 i) – (4 + 5· i) = –1 – i

==

6. TRIGONOMETRIC FORM OF 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.

Let's consider trigonometric form writing a complex number. Real and imaginary parts of a complex number Z=A+B· i are expressed through 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 =– modulus of a complex number.

The number j is called argument of a complex number.

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

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

As mentioned above = r =, equality (2) can be written in the form

A+B i =· cosj + i · sinj, from where, equating the real and imaginary parts, we get:

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 formula (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 MODULE AND ARGUMENT OF A COMPLEX NUMBER

Using the trigonometric form it is convenient to find the product and 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 using the formula:

Z 1 Z 2 = r 1 · r 2 (5)

From formula (5) it follows that When multiplying complex numbers, their modules are multiplied and their arguments are added.

If Z 1 =Z 2 then we get:

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

Z 3 =Z 2 Z= 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 Moivre's formula.

The quotient of two complex numbers written in trigonometric form can be found using 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 = ; Z2 = –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

Z 1 = cos0+ i× sin0 = 1 + 0 = 1

k = 1

Z 2 = cos+ i× sin = 0 + i = i

k = 2

Z 3 = cosp + i sinp = –1 + 0 = –1

k = 3

Z 4 = cos+ i× sin

Answer: Z 13 = 1

Z 24 = i

8. RAISING TO A POWER AND EXTRACTING THE ROOT

From formula 6 it is clear that raising the complex number r· (cosj + i sinj ) to a positive integer power with a natural exponent, its module 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 sinnj)

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

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

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

The equation Z n = w will take the form:

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

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

So, all solutions can be written as follows:

Z K =, kО Z (8)

Formula 8 is called Moivre's second formula.

Thus, if w 0, then there are exactly n roots of degree n from the number w: they are all contained in formula 8. All roots of degree n from the number w have the same module , but different arguments, differing by a term that is a multiple of the number . It follows that complex numbers, which are roots of degree n from a complex number w, correspond to 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 a clear meaning. Therefore, when using it, you should clearly understand what is meant by this symbol. For example, when using the notation , you should consider making it clear whether this symbol means a pair of complex numbers i And –i , or one thing, then which one.

Equations of higher degrees

Formula 8 determines all the roots of a binomial 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 proven: every algebraic equation has at least one root in the set of complex numbers. This theorem was proven by the German mathematician Carl Gauss in 1779.

Based on Gauss's theorem, it can be proven 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

It follows 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 stated give solutions to 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 equations of the third and fourth degrees the formulas are cumbersome, and for equations of degree above the fourth such formulas do not exist at all. Absence general method It doesn’t hurt to find 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 free term.

Let's prove this theorem:

Let Z = k be the 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 k is a divisor of the number a 0 .

9.QUADRATE EQUATION WITH COMPLEX UNKNOWN

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

This is the equation:

Let's 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 of 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 using formula 10. Moreover, if the discriminant, i.e. radical expression in formula 10

D = b 2 – 4× a× c

is positive, then the equation a× Z 2 + b× Z + c = 0 are 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.

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

Let us 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 following properties hold:

Z 1× Z 2 =

1. For all complex Z the formula is valid

a× Z 2 + b× Z + c = a× (Z – Z 1)× (Z – Z 2)

Example 5:

Z 2 – 6 Z + 10 = 0

D = b 2 – 4 a c

D = 6 2 – 4 10 = – 4

– 4 = i 2 ·4

Z 1.2 =

Answer: Z 1 = Z 2 = 3 + i

Example 6:

3 Z 2 +2 Z + 1 = 0

D = b 2 – 4 a c

D = 4 – 12 = – 8

D = –1·8 = 8· i 2

Z 1.2 = =

Answer: Z 1 = Z 2 = –

Example 7:

Z 4 – 8 Z 2 – 9 = 0

t 2 – 8 t – 9 = 0

D = b 2 – 4 a c = 64 + 36 = 100

t 1 = 9 t 2 = – 1

Z 2 = 9 Z 2 = – 1

Z 3.4 = i

Answer: Z 1.2 =3, Z 3.4 = i

Example 8:

Z 4 + 2 Z 2 – 15 = 0

t 2 + 2 t – 15 = 0

D = b 2 – 4 a c = 4 + 60 = 64

t 1.2 = = = –14

t 1 = – 5 t 2 = 3

Z 2 = – 5 Z 2 = 3

Z 2 = – 1·5 Z 3.4 =

Z 2 = i 2 ·5

Z 1.2 = i

Answer: Z 1.2 = i , Z 3.4 =

Example 9:

Z 2 = 24 10 i

Let Z = X + Y i

(X + Y i ) 2 = X 2 + 2· X· Y· i – Y2

X 2 + 2 X Y i – Y 2 = 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 = – 25

t 1 = 25 t 2 = – 1

X 2 = 25 X 2 = – 1 - no solutions

X 1 = 5 X 2 = – 5

Y 1 = – Y 2 =

Y 1 = – 1 Y 2 = 1

Z 1.2 =(5 – i )

Answer: Z 1.2 =(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 = 0 / –1

Y 2 – 2Y + 2 = 0

D = b 2 – 4 a c = 4 – 8 = – 4

– 4 = – 1·4 = 4· i 2

Y 1.2 = = = 1 i

Y 1 = 1– i Y 2 = 1 + i

X 1 = 1 + i X 2 = 1– i

Answer: (1 + i ; 1–i }

{1–i ; 1 + i }

Let's square it